On the Green Function for the Anisotropic Simple Cubic Lattice

Annals of Physics 291, 71–133 (2001) doi:10.1006/aphy.2001.6148, available online at http://www.idealibrary.com on

On the Green Function for the Anisotropic Simple Cubic Lattice R. T. Delves and G. S. Joyce Wheatstone Physics Laboratory, King’s College, University of London, Strand, London WC2R 2LS, United Kingdom Received December 5, 2000

The analytical properties of the lattice Green function Z πZ πZ π 1 dθ1 dθ2 dθ3 G(α, w) = 3 π 0 0 0 w − cos θ1 − cos θ2 − α cos θ3 are investigated, where w = u + iv is a complex variable in the (u, v) plane and α is a real parameter in the interval (0, ∞). In particular, it is shown that the function yG (α, z) ≡ wG(α, w), where z = 1/w2 , is a solution of a fourth-order linear differential equation of the type 4 X

f j (α, z)D4− j y = 0,

j=0

where f j (α, z) is a polynomial in the variables α and z and D ≡ d/dz. It is then proved that the solutions of this differential equation can all be expressed in terms of a product of two functions H1 (α, z) and H2 (α, z) which satisfy second-order linear differential equations of the normal type [D2 + U+ (α, z)]y = 0, [D2 + U− (α, z)]y = 0, respectively, where U± (α, z) are complicated algebraic functions of α and z. Next Schwarzian transformation theory is used to reduce both these second-order differential equations to the standard Gauss hypergeometric differential equation. From this result it is deduced that · ¸· ¸ 2 2 2 p wG(α, w) = p K (k+ ) K (k− ) , π 1 − (2 − α)2 z + 1 − (2 + α)2 z π where i−3 p 1 hp 1 − 1 − (2 − α)2 z + 1 − (2 + α)2 z 2 2 ·q ¸ q q q √ √ √ √ × 1 + (2 − α) z 1 − (2 + α) z + 1 − (2 − α) z 1 + (2 + α) z

2 2 k± ≡ k± (α, z) =

( ×

±16z +

"q q p √ √ 1 − α2 z 1 + (2 − α) z 1 + (2 + α) z

¸ ) q q √ √ 2 + 1 − (2 − α) z 1 − (2 + α) z

and K (k) denotes the complete elliptic integral of the first kind with a modulus k. This basic formula is valid for all values of w = u + iv which lie in the (u, v) plane, provided that a cut is made along the real axis from w = −2 − α to w = 2 + α. In the remainder of the paper exact series expansions for G(α, w) 71 0003-4916/01 $35.00 C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.

72

DELVES AND JOYCE are derived which are valid in a sufficiently small neighbourhood of the branch-point singularities at w = 2 + α, w = α, and w = |2 − α|. In all cases it is shown that the real and imaginary parts of the coefficients in the analytic part of these expansions can be expressed in terms of complete elliptic integrals of the first and second kinds, while the coefficients in the singular part of the expansions can be expressed in terms of rational functions of α. The behaviour of G(α, w) in the immediate neighbourhood of w = 0 is also investigated in a similar manner. Finally, several applications of the results are made in lattice statistics. °C 2001 Academic Press

1. INTRODUCTION The lattice Green function 1 G(l, m, n; α, w) = 3 π

Z πZ πZ 0

0

π 0

cos lθ1 cos mθ2 cos nθ3 dθ1 dθ2 dθ3 , w − cos θ1 − cos θ2 − α cos θ3

(1.1)

where {l, m, n} denotes a set of integers, w = u + iv is a complex variable in the (u, v) plane, and α is a real parameter in the interval −∞ < α < ∞, plays an important role in many lattice statistical problems which involve the simple cubic lattice with partially anisotropic nearest-neighbour interactions [1–8]. We can assume, without loss of generality, that l ≥ m ≥ 0 and n ≥ 0. When α = 0 the Green function (1.1) reduces to G(l, m, n; 0, w) = G sq (l, m; w)δn,0 ,

(1.2)

where δn,m denotes a Kronecker delta function and G sq (l, m; w) is the Green function for the twodimensional square lattice with isotropic nearest-neighbour interactions [9]. It is also readily seen from (1.1) that G(l, m, n; −α, w) = (−1)n G(l, m, n; α, w).

(1.3)

We shall, therefore, restrict our attention to the case 0 < α < ∞. For the case l = m = n = 0 we shall make use of the more compact notation G(α, w) ≡ G(0, 0, 0; α, w).

(1.4)

The triple integral (1.1) defines a single-valued analytic function G(l, m, n; α, w) in the complex (u, v) plane provided that a cut is made along the real axis from w = −2 − α to w = 2 + α, where α > 0. We find from (1.1) that G(l, m, n; α, w) satisfies the symmetry relation G(l, m, n; α, −w) = (−1)l+m+n+1 G(l, m, n; α, w).

(1.5)

In many applications one requires the limiting behaviour of G(l, m, n; α, w) as w approaches the real u axis. It is convenient, therefore, to introduce the further definitions G ± (l, m, n; α, u) ≡ lim G(l, m, n; α, u ± i²) ≡ G R (l, m, n; α, u) ∓ iG I (l, m, n; α, u), ²→0+

(1.6)

where −∞ < u < ∞ and ² is an infinitesimal positive number. For the special case l = m = n = 0 we shall write (1.6) in the simpler form G ± (α, u) ≡ lim G(α, u ± i²) ≡ G R (α, u) ∓ iG I (α, u). ²→0+

(1.7)

ON THE SIMPLE CUBIC GREEN FUNCTION

73

We find using (1.5) that the real part G R is an odd or even function of u and the imaginary part G I is an even or odd function, according to whether l + m + n is even or odd. When |u| ≥ 2 + α the imaginary part of G ± (l, m, n; α, u) is always equal to zero. It can also be shown [10] that on the cut the function G ± (l, m, n; α, u) exhibits branch-point singularities at u = ±α, −2 ± α, and 2 ± α. A simple integral representation for the Green function (1.6) can be derived by first applying the formula Z ∞ exp[±i(λ ± i²)t] dt = (λ ± i²)−1 , (1.8) ∓i 0

to the denominator of the integrand in (1.1) with w = u ± i², where λ is real and ² > 0. The resulting multiple integral can then be simplified using the standard result 1 π

Z

π

cos(nθ ) exp(it cos θ ) dθ = in Jn (t),

(1.9)

0

where Jn (t) denotes a Bessel function of the first kind of order n. In this manner, we find that [11] ±

Z

∞

G (l, m, n; α, u) = (∓i)

l+m+n+1

exp(±iut)Jl (t)Jm (t)Jn (αt) dt,

(1.10)

0

where −∞ < u < ∞. Maradudin et al. [6] have also shown that Z G(l, m, n; α, w) =

∞

exp(−wt)Il (t)Im (t)In (αt) dt,

(1.11)

0

where In (t) denotes a modified Bessel function of the first kind and |w| ≥ 2 + α with α > 0. When l = m = n = 0, Eqs. (1.7) and (1.10) give the formulae Z G R (α, u) =

∞

0

Z G I (α, u) =

∞

0

sin(ut)J02 (t)J0 (αt) dt,

(1.12)

cos(ut)J02 (t)J0 (αt) dt,

(1.13)

where −∞ < u < ∞. More generally, it is possible to express the Green function G(α, w) in the form Z G(α, w) =

2+α

−2−α

ρ(α, u) du, w−u

(1.14)

where w is any point in the cut (u, v) plane and ρ(α, u) =

1 G I (α, u) π

(1.15)

is a density-of-states function [8, 11]. A series representation for G(α, w) can now be obtained by expanding the integrand in (1.14) in powers of 1/w. We find that G(α, w) =

∞ 1 X 1 µ2n (α) 2n , w n=0 w

(1.16)

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DELVES AND JOYCE

where |w| ≥ 2 + α and Z µ2n (α) = 2

2+α

u 2n ρ(α, u) du.

(1.17)

0

It should be noted that the odd moments of the density function ρ(α, u) are zero because ρ(α, u) is an even function of u. Most of the work on the exact closed-form evaluation of the Green function (1.1) has been carried out for the isotropic case α = 1. For the special case α = 1 and w = 3, Watson [12] proved that ¸2 · √ √ ´ 2 √ K (k) , G(0, 0, 0; 1, 3) ≡ G(1, 3) = 18 + 12 2 − 10 3 − 7 6 π ³

(1.18)

where ³ √ ´ √ ´³√ 3− 2 k = 2− 3

(1.19)

and K (k) denotes the complete elliptic integral of the first kind with a modulus k. It is interesting to note that the modulus k in (1.19) is equal to the singular value k[6] (see [13, 14]). For any positive integer N the singular value k[N ] has the property that K 0 (k[N ]) √ = N, K (k[N ])

(1.20)

where K 0 (k) is the complementary complete elliptic integral of the first kind. An exact formula for the more general Green function G(1, w) was first derived by Joyce [15, 16]. In particular, it was found that µ wG(1, w) = (1 − η)

1 2

1 1− η 4

¶ 12 ·

¸·

2 K (k+ ) π

¸ 2 K (k− ) , π

(1.21)

where r ¸ · µ ¶ 1 1 1 p = 1−η , 1±η 1− η− 1− η 2 4 2 ´−2 ³√ √ η = −16z 1 − z + 1 − 9z

2 k±

(1.22) (1.23)

and z = 1/w 2 . This result can be used to calculate G(1, w) at any point w in the (u, v) plane provided that a cut is made along the real axis from w = −3 to w = +3. It was shown by Morita [17] that the Green function G(l, m, n; 1, w) at an arbitrary lattice site (l, m, n) could be expressed in terms of G(0, 0, 0; 1, w), G(2, 0, 0; 1, w), and G(3, 0, 0; 1, w) by using a coupled set of recurrence relations, while Horiguchi and Morita [18] demonstrated that the Joyce formula (1.21) could be used to evaluate G(2, 0, 0; 1, w) and G(3, 0, 0; 1, w) in closed forms which involve sums of products of the complete elliptic integrals of the first and second kinds. From these results it follows that it is always possible, at least in principle, to evaluate G(l, m, n; 1, w) in terms of the complete elliptic integrals K (k± ) and E(k± ). More recently, Joyce [19, 20] has proved that the moduli k± = k± (η) in (1.22) are related by the cubic modular transformation of order 3 (see [13, p. 102]). This remarkable connection enables one

ON THE SIMPLE CUBIC GREEN FUNCTION

75

to write (1.21) in the following much simpler parametric form, wG(1, w) =

· ¸2 2 (1 − 9ξ 4 ) ˜ K ( k) , (1 − ξ )3 (1 + 3ξ ) π

(1.24)

where 16ξ 3 , (1 − ξ )3 (1 + 3ξ ) ´− 12 ³ ´12 ³ √ √ 1− 1−z , ξ = 1 + 1 − 9z

k˜ 2 =

(1.25) (1.26)

and z = 1/w2 . If the formula (1.24) is combined with the earlier work of Morita [17] and Horiguchi and Morita [18] one finds that (Joyce, unpublished work) ·

l+m+n

(3/w)

¸2 2 ˜ wG(l, m, n; 1, w) = R0 (l, m, n; ξ ) + R1 (l, m, n; ξ ) K (k) + R2 (l, m, n; ξ ) π · ¸· ¸ ¸ · 2 ˜ 2 2 ˜ 2 ˜ K (k) E (k) + R3 (l, m, n; ξ ) E (k) , (1.27) × π π π

where {R j (l, m, n; ξ ) : j = 0, 1, 2, 3} denotes a set of rational functions of ξ and the modulus k˜ is defined in (1.25). For example, when l = 2, m = 1, n = 0 it can be shown that 9 − (1 + ξ )(1 − 3ξ )(1 − ξ )−2 8 ¸2 · 2 ˜ K (k) × (1 − 9ξ 4 )−2 (15 − 46ξ 2 − 306ξ 6 + 81ξ 8 ) π ¸· ¸ · 2 ˜ 27 2 ˜ K (k) E (k) + (7 − 50ξ 2 + 90ξ 6 + 81ξ 8 )(1 − 9ξ 4 )−2 4 π π ¸2 · 243 −1 2 ˜ , (1 − ξ )3 (1 + 3ξ )(1 − 9ξ 4 ) E (k) (1.28) − 8 π

(3/w)3 wG(2, 1, 0; 1, w) = 36ξ 2 (1 − ξ 2 )(1 − 9ξ 2 )(1 − 9ξ 4 )

−2

˜ denotes a complete elliptic integral of the second kind. For the special case w = 3 the where E (k) formula (1.27) can be reduced to G(l, m, n; 1, 3) = r0 (l, m, n) + r1 (l, m, n)G(1, 3) + r2 (l, m, n)[π 2 G(1, 3)]−1 ,

(1.29)

where {r j (l, m, n) : j = 0, 1, 2} denotes a set of rational numbers and G(1, 3) is given by the Watson formula (1.18). In particular, we find from (1.28) that G(2, 1, 0; 1, 3) =

9 1 3 + G(1, 3) − [π 2 G(1, 3)]−1 . 3 8 4

(1.30)

The result (1.29) has also been derived independently by Glasser and Boersma [21]. Montroll [22] extended the work of Watson [12] on G(1, 3) and established an exact closed-form expression for the more general anisotropic Green function G(α, 2 + α), where 0 < α < ∞. His

76

DELVES AND JOYCE

final result can be written as

√ ³ ´ 4 √ 2 √ √ 2 1+α− 2+α K [k+ (α)]K [k− (α)], G(α, 2 + α) = α π2

(1.31)

where k± (α) =

´³√ √ ´ √ 1 ³√ √ 2 1+α− 2+α 2+α± 2 . α

(1.32)

When α = 1 we can simplify (1.31) using the transformation formula µ ¶ 12 3 K [k+ (1)] = [1 + k− (1)]K [k− (1)]. 2

(1.33)

In this manner, we recover the Watson formula (1.18). It appears that (1.31) is the only exact elliptic function formula currently available in the literature for the Green function (1.1) which is valid for general values of the anisotropy parameter α. However, Abe and Katsura [10] have used (1.10) and Mellin–Barnes integral methods to prove that, for arbitrary values of α and u in the (α, u) plane, the Green function G ± (l, m, n; α, u) can always be expressed in terms of various double series associated with the Kamp´e de F´eriet function. Our main aim in this paper is to give a detailed account of the analytic properties of the lattice Green function G(α, w). In Sections 2–4 we investigate various linear differential equations which are associated with G(α, w). Hence, we deduce in Section 5 that G(α, w) can be expressed in terms of a product of two complete elliptic integrals of the first kind. It is shown that this key result enables one to calculate the value of G(α, w) for any w = u + iv in the cut (u, v) plane and for any α ∈ (0, ∞). In Sections 7–10 the exact behaviour of G(α, w) in a sufficiently small neighbourhood of the points w = 2 + α, α, |2 − α| and 0 is established. Finally, some applications of the results in lattice statistics are discussed in Section 11. 2. BASIC RESULTS In this section we shall derive a recurrence relation for the even moments (1.17) and a fourth-order differential equation for the Green function G(α, w). (a) Moment Expansion for G(α, w) about w = ∞ We begin by expanding the integrand in (1.1) with l = m = n = 0 in powers of 1/w and integrating the resulting series term by term. If this expansion is compared with (1.16) we see that wG(α, w) ≡ yG (α, z) =

∞ X

µ2n (α)z n ,

(2.1)

n=0

where z = 1/w 2 and 1 µ2n (α) = 3 π

Z πZ πZ 0

0

π

(cos θ1 + cos θ2 + α cos θ3 )2n dθ1 dθ2 dθ3 .

(2.2)

0

The series (2.1) is convergent provided that |z| ≤ 1/(2 + α)2 . An explicit expression for µ2n (α) can be derived from (2.2) in terms of a terminating generalised hypergeometric series. The final result is ¡1¢ µ2n (α) =

2 n

(1)n



−n,

 α 2n 3 F2 

1,

−n, 1;

1 ; 2

  4/α 2  ,

(2.3)

ON THE SIMPLE CUBIC GREEN FUNCTION

77

where (b)n denotes the Pochhammer symbol. For the isotropic case α = 1 the formula (2.3) is in agreement with the earlier work of Joyce [16]. From (2.3) one finds that µ0 (α) = 1, µ2 (α) = µ4 (α) = µ6 (α) = µ8 (α) = µ10 (α) =

(2.4)

1 (2 + α 2 ), 2 3 (6 + 8α 2 + α 4 ), 8 5 (20 + 54α 2 + 18α 4 + α 6 ), 16 35 (70 + 320α 2 + 216α 4 + 32α 6 + α 8 ), 128 63 (252 + 1750α 2 + 2000α 4 + 600α 6 + 50α 8 + α 10 ). 256

(2.5) (2.6) (2.7) (2.8) (2.9)

In order to investigate the properties of the moments {µ2n (α) : n = 0, 1, 2, . . .} we introduce the exponential generating function Z πZ πZ π 1 exp[t(cos θ1 + cos θ2 + α cos θ3 )] dθ1 dθ2 dθ3 , (2.10) E 0 (t) = 3 π 0 0 0 =

∞ X µ2n (α) n=0

(2n)!

t 2n ,

|t| < ∞.

The application of the standard integral representation for the modified Bessel function Z 1 π exp(t cos θ) dθ I0 (t) = π 0

(2.11)

(2.12)

to Eq. (2.10) gives the simple formula E 0 (t) = f (t)g(t),

(2.13)

where f (t) = [I0 (t)]2 and g(t) = I0 (αt). (b) Differential Equation for E 0 (t) We can derive a differential equation for the exponential generating function E 0 (t) by differentiating (2.13) n = 1, 2, . . . times. If the resulting expressions are simplified using the differential relations t 2 f 000 (t) = −3t f 00 (t) + (4t 2 − 1) f 0 (t) + 4t f (t), 00

0

tg (t) = −g (t) + α t g(t), 2

(2.14) (2.15)

it is found that t n−1

5 X dn E 0 = dn, j (α, t)E j (t), dt n j=0

(2.16)

where E 1 (t) = f (t)g 0 (t), E 2 (t) = f 0 (t)g(t), E 3 (t) = f 0 (t)g 0 (t), E 4 (t) = f 00 (t)g(t), E 5 (t) = f 00 (t)g 0 (t), and dn, j (α, t) is a polynomial in the variables α 2 and t 2 . (It should be noted that for n = 1, 2 not all the terms are present in the summation (2.16).)

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DELVES AND JOYCE

Next we use the set of linear equations formed from (2.16) with n = 1, . . . , 5 to obtain formulae for {E j (t) : j = 1, . . . , 5} in terms of E 0 (t) and its derivatives E 0(m) (t), where m = 1, . . . , 5. If these results are substituted in Eq. (2.16) with n = 6 we find that E 0 (t) is a solution of the following sixth-order differential equation: £ ¤ d6 E 0 £ ¤ 5 £ 4 2 2 d E0 t 5 3 − 4(4 − α 2 )t 2 + 3t )t + t 3 87 − (248 − 47α 2 )t 2 11 − 12(4 − α 6 5 dt dt 4 ¤ £ ¤ 3 E d 0 2 2 2 2 2 4 d E0 + 4(4 − α 2 )(8 + 3α 2 )t 4 + 2t )t + 12(4 − α )(8 + 3α )t 21 − (44 + 49α dt 4 dt 3 £ ¤ d2 E 0 − t 3 + (376 + 119α 2 )t 2 − (752 + 64α 2 − 51α 4 )t 4 + 4(4 − α 2 )(16 + 3α 4 )t 6 dt 2 £ ¤ dE 0 + 3 − (152 + 25α 2 )t 2 + (80 − 56α 2 + 69α 4 )t 4 − 12(4 − α 2 )(16 + 3α 4 )t 6 dt £ ¤ 3 2 4 2 2 4 2 2 2 3 4 (2.17) + t 24(10 + α + α ) − (4 − α )(64 + 28α + α )t + 4α (4 − α ) t E 0 = 0. The complicated manipulations involved in the derivation of (2.17) were carried out using a computer algebra system. (c) Recurrence Relation for µ2n (α) and a Differential Equation for yG (α, z) It is now possible to determine a recurrence relation for µ2n (α) by substituting the series (2.11) in the differential equation (2.17). The final result is £ 24n(n + 1)3 µ2n+2 (α) − 4n(2n + 1) 3(2 + α 2 ) + 10(2 + α 2 )n + 3(56 − 9α 2 )n 2 − 48(4 − α 2 )n 3 £ ¤ + 16(4 − α 2 )n 4 µ2n (α) + 2(4n 2 − 1) (380 + 34α 2 − 33α 4 ) − 2(4 − α 2 )(214 + 75α 2 )n ¤ + 3(976 + 112α 2 − 85α 4 )n 2 − 64(4 − α 2 )(8 + 3α 2 )n 3 + 16(4 − α 2 )(8 + 3α 2 )n 4 µ2n−2 (α) £ − (4 − α 2 )(n − 1)(2n − 3)(4n 2 − 1) (576 + 28α 2 + 97α 4 ) − 48(16 + 3α 4 )n + 16(16 ¤ + 3α 4 )n 2 µ2n−4 (α) + 8α 2 (4 − α 2 )3 (n − 1)(n − 2)(2n − 3)(2n − 5)(4n 2 − 1)µ2n−6 (α) = 0, (2.18) where n = 1, 2, . . . , with the initial conditions µ0 (α) = 1 and µ2 (α) = 12 (2 + α 2 ). We can use the recurrence relation (2.18) and the series (2.1) to show that the modified Green function yG (α, z) ≡ wG(α, w) is a solution of the sixth-order differential equation L6 (y) = 0, where £ ¤£ ¤ L6 ≡ 256(4 − α 2 )z 5 (1 − α 2 z) 1 − (2 − α)2 z 1 − (2 + α)2 z D6 £ ¤ + 128(4 − α 2 )z 4 25 − 34(8 + 3α 2 )z + 43(16 + 3α 4 )z 2 − 52α 2 (4 − α 2 )2 z 3 D5 £ − 16z 2 3 − (2632 − 643α 2 )z + (43760 + 5440α 2 − 4083α 4 )z 2 ¤ − (4 − α 2 )(37376 + 28α 2 + 6997α 4 )z 3 + 3560α 2 (4 − α 2 )3 z 4 D4 £ − 8z 30 − (4592 − 923α 2 )z + 8(18220 + 2213α 2 − 1662α 4 )z 2 ¤ − (4 − α 2 )(188288 + 644α 2 + 35051α 4 )z 3 + 24080α 2 (4 − α 2 )3 z 4 D3

(2.19)

79

ON THE SIMPLE CUBIC GREEN FUNCTION

£ − 4 48 − 2(832 + 161α 2 )z + (128960 + 14104α 2 − 10401α 4 )z 2

¤ − 15(4 − α 2 )(21184 + 252α 2 + 3873α 4 )z 3 + 60900α 2 (4 − α 2 )3 z 4 D2 £ + 6 24(11 + 3α 2 ) − 2(2320 + 44α 2 + 69α 4 )z + 5(4 − α 2 )(8000 + 364α 2 + 1357α 4 )z 2 ¤ £ −14560α 2 (4 − α 2 )3 z 3 D − 6 12(10 + α 2 + α 4 ) ¤ (2.20) − 5(4 − α 2 )(64 + 28α 2 + α 4 )z + 560α 2 (4 − α 2 )3 z 2 ,

D ≡ d/dz, and z = 1/w2 . (d) Reduction of the Order of the Differential Equation L6 (y) = 0 The differential equation (2.19) has singular points in the z plane at z = 0, z 1 = 1/(2 + α)2 ,

(2.21)

z 2 = 1/α ,

(2.22)

z 3 = 1/(2 − α)2 ,

(2.23)

2

and z = ∞. In the neighbourhood of the points {z m : m = 1, 2, 3} and z = ∞ all the integrals of (2.19) are of the regular type (see [23, p. 74]). However, in the vicinity of the singular point z = 0 there are only four independent regular integrals of (2.19). It is clear from the convergent series (2.1) that we can take one of these regular integrals to be the modified Green function yG (α, z). Under these circumstances, it can be shown [23, p. 223] that the regular integrals at z = 0 form a fundamental system for a differential equation of the Fuchsian type L4 (y) = 0,

(2.24)

where L4 denotes a linear fourth-order differential operator. It is also possible to express the operator (2.20) in the product form [23, p. 228] L6 = L2 L4 ,

(2.25)

where L2 is a linear differential operator of second order. We have constructed the reduced operator L4 by fitting the moment expansion (2.1) to a general linear fourth-order differential equation with polynomial coefficients. It is then possible to determine the operator L2 using a method described by Ince [24, p. 420]. In this manner, it is found that L2 =

© 1 32(4 − α 2 )z 3 [3 + 5(4 − α 2 )z] D2 2 2 [3 + 5(4 − α )z]

ª

+ 16(4 − α 2 )z 2 [21 + 25(4 − α 2 )z]D − 6[3 + (4 − α 2 )z]

(2.26)

and L4 =

4 X

f j (α, z)D4− j ,

(2.27)

j=0

where £ ¤£ ¤£ ¤ f 0 (α, z) = 8z 2 (1 − α 2 z) 1 − (2 − α)2 z 1 − (2 + α)2 z 3 + 5(4 − α 2 )z ,

(2.28)

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DELVES AND JOYCE

£ f 1 (α, z) = 20z 6 − 5(8 + 7α 2 )z − (224 + 52α 2 − 75α 4 )z 2

¤ + 3(4 − α 2 )(96 − 20α 2 + 23α 4 )z 3 − 23α 2 (4 − α 2 )3 z 4 , £ f 2 (α, z) = 96 − 4(544 + 233α 2 )z − 2(448 + 608α 2 − 1365α 4 )z 2

(2.29)

¤ + 4(4 − α 2 )(3040 − 974α 2 + 811α 4 )z 3 − 1350α 2 (4 − α 2 )3 z 4 , (2.30) £ f 3 (α, z) = −72(11 + 3α 2 ) + 6(736 + 8α 2 + 177α 4 )z ¤ (2.31) + 3(4 − α 2 )(1760 − 1076α 2 + 589α 4 )z 2 − 975α 2 (4 − α 2 )3 z 3 , £ ¤ f 4 (α, z) = 36(10 + α 2 + α 4 ) + 3(4 − α 2 )(40 − 104α 2 + 31α 4 )z − 75α 2 (4 − α 2 )3 z 2 . (2.32) It has been verified explicitly that (2.26) and (2.27) satisfy the operator relation (2.25). From the above rigorous analysis we deduce that the modified Green function yG (α, z) is a solution of the fourth-order differential equation L4 (y) = 0, where L4 is defined in Eqs. (2.27)– (2.32). This differential equation is of the Fuchsian type with six regular singular points at z = 0, {z m : m = 1, 2, 3}, z4 =

3 2 (α − 4)−1 , 5

(2.33)

and ∞. The Riemann P-symbol [24, p. 370] associated with the equation L4 (y) = 0 is given by 

0

 0   0 P   0  1

z1

z2

z3

z4

∞

0

0

0

0

1

1

1

1

1

1 2

2

2

2

2

3 2

1 2

1 2

1 2

4

5 2

     z .    

(2.34)

In this scheme the singular points are placed on the first row with the roots of the corresponding indicial equations beneath them. For an arbitrary nth order Fuchsian equation with ν regular singular points in the finite z plane and a regular singular point at z = ∞, it can be shown [24, p. 371] that the sum of all the exponents in the Riemannian scheme is an invariant equal to 12 n(n − 1)(ν − 1). We see directly from (2.34) that the differential equation L4 (y) = 0 has the correct Fuchsian invariant of 24. The singular point z 4 is particularly interesting because the general solution of L4 (y) = 0 is analytic at z = z 4 . This unusual type of regular singular point is known as √ an apparent (or accidental) singularity [24, p. 406]. It should be noted that when α = 1, 2, 8, and 10 it is necessary to alter the structure of the P-symbol (2.34) because at least two of the singular points z = 0, {z m : m = 1, 2, 3, 4}, and ∞ are confluent. Finally, we derive a further recurrence relation for µ2n (α) by substituting the moment expansion (2.1) in the reduced differential equation L4 (y) = 0. We find that £ ¤ 24n(n + 1)3 µ2n+2 (α) − 4n 3(2 + α 2 ) + 16(2 + α 2 )n + (152 + 7α 2 )n 2 + 4(2 + 7α 2 )n 3 µ2n (α) £ + 2(2n − 1) (236 + 22α 2 − 21α 4 ) − 12(70 + 7α 2 − 6α 4 )n + 3(336 + 40α 2 − 27α 4 )n 2 £ ¤ − 8(28 + 5α 2 − 6α 4 )n 3 µ2n−2 (α) + (4 − α 2 )(2n − 1)(2n − 3) (360 + 16α 2 + 61α 4 ) ¤ − 3(160 + 4α 2 + 29α 4 )n + 4(40 − 6α 2 + 9α 4 )n 2 µ2n−4 (α) − 5α 2 (4 − α 2 )3 (n − 2)(2n − 1)(2n − 3)(2n − 5)µ2n−6 (α) = 0,

(2.35)

ON THE SIMPLE CUBIC GREEN FUNCTION

81

where n = 1, 2, . . . , with the initial conditions µ−4 (α) = µ−2 (α) ≡ 0, µ0 (α) = 1, and µ2 (α) = 1 (2 + α 2 ). This recurrence relation can be considered to be simpler than the relation (2.18) because 2 it involves polynomial coefficients which are all of degree 4 in the variable n. (e) Fourth-Order Differential Equation for G(α, w). If the transformations z = 1/w2 and y = wG are applied to the differential equation L4 (y) = 0 it is found that the Green function G(α, w) is a solution of the fourth-order differential equation 4 X j=0

g j (α, w)

d4− j G = 0, dw4− j

(2.36)

where £ £ ¤ ¤£ ¤ g0 (α, w) = w w2 − (2 + α)2 (w2 − α 2 ) w 2 − (2 − α)2 5(4 − α 2 ) + 3w 2 , £ g1 (α, w) = 5α 2 (4 − α 2 )3 + (4 − α 2 )(320 + 36α 2 + 51α 4 )w2 ¤ − 3(448 + 60α 2 − 51α 4 )w4 + 7(16 − 19α 2 )w6 + 36w 8 , £ g2 (α, w) = w 20(4 − α 2 )(4 − 2α 2 + α 4 ) − (2800 + 308α 2 − 249α 4 )w2 ¤ + 68(11 − 5α 2 )w4 + 111w 6 , £ g3 (α, w) = − 20(4 − α 2 )(4 − 2α 2 + α 4 ) + 4(236 + 22α 2 − 21α 4 )w2 ¤ − (836 − 245α 2 )w4 − 87w 6 , £ ¤ g4 (α, w) = 9w3 3(4 − α 2 ) + w 2 .

(2.37)

(2.38)

(2.39)

(2.40) (2.41)

This result will be used in Sections 7–9 to investigate the singular behaviour of G(α, w) in the w plane. We have also derived the differential equation (2.36) by following an alternative method developed by Iwata [25].

3. REPRESENTATION OF THE SOLUTIONS OF L4 (y) = 0 AS A PRODUCT OF SOLUTIONS OF TWO SECOND-ORDER DIFFERENTIAL EQUATIONS Our main task in this section is to show that the solutions of L4 (y) = 0, where L4 is defined in (2.27), can all be expressed in the product form " µ #µ ¶ ¶ 3 z 1/2 z −1/2 1 Y H1 (α, z)H2 (α, z), 1− 1− y(α, z) = z i=1 zi z5

(3.1)

where {z i : i = 1, 2, 3} are the regular singular points defined in (2.21)–(2.23), z 5 = 3/(4 − α 2 ),

(3.2)

and H1 (α, z) and H2 (α, z) are solutions of two second-order differential equations in normal form.

82

DELVES AND JOYCE

(a) Proof of the Product Form (3.1) We begin the analysis by writing any solution of L4 (y) = 0 as y(α, z) = T (α, z)H (α, z),

(3.3)

where T (α, z) = z

β0

" µ 3 Y i=1

z 1− zi

¶βi #µ

z 1− z5

¶γ1

.

(3.4)

The exponents β0 , β1 , β2 , β3 , γ1 in (3.4) are initially undetermined constants. If the factorization (3.1) is to be obtained it is also necessary to include an extra singularity in (3.4) whose position z 5 is to be determined. The values of the parameters {z 5 ; β0 , β1 , β2 , β3 , γ1 } will be fixed at a later stage by the stringent requirements of the product formula (3.1). If we make the substitution (3.3) in L 4 (y) = 0 we find that " D4 +

4 X

# h j (α, z)D4− j H (α, z) = 0,

(3.5)

¶ j µ X m + 4 − j f j−m (α, z) Dm T (α, z) , f 0 (α, z) T (α, z) 4− j m=0

(3.6)

j=1

where h j (α, z) =

¡¢ D ≡ d/dz, nr is the binomial coefficient, and the coefficients { f i (α, z) : i = 0, 1, . . . , 4} are defined in (2.28)–(2.32). If the formula (3.4) is applied to (3.6) it is seen that the coefficient h j (α, z) is expressible as a ratio of two polynomials in the variables α, z with coefficients which depend on the undetermined parameter set {z 5 ; β0 , β1 , β2 , β3 , γ1 }. Next we investigate the possibility that the function H (α, z) can be written in the product form H (α, z) = H1 (α, z)H2 (α, z),

(3.7)

where H1 (α, z) and H2 (α, z) are solutions of the differential equations [D2 + U+ (α, z)]y = 0

(3.8)

[D2 + U− (α, z)]y = 0,

(3.9)

and

respectively, and U± (α, z) are functions of α and z which are to be determined. It follows from (3.8) and (3.9) that H (α, z) must also satisfy the fourth-order differential equation (see [26; 27, p. 146]) ¸ D3 H + 2(U+ + U− )DH + H D(U+ + U− ) + (U+ − U− )H = 0, D U+ − U− ·

(3.10)

provided that U+ 6= U− . We anticipate that the functions U± = U± (α, z) will both involve a square

ON THE SIMPLE CUBIC GREEN FUNCTION

83

root function, and it is convenient to write U± in the alternative form U± =

p ´ 1³ V1 ± V2 . 2

(3.11)

If the formula (3.11) is applied to (3.10) we find that the product form (3.7) is a solution of the differential equation µ

· D − 4

µ ¶ ¶ µ ¶¸ 0 V20 V20 3 2 0 00 0 V2 + V2 H (α, z) = 0, D + 2V1 D + 3V1 − V1 D + V1 − V1 2V2 V2 2V2

(3.12)

where Vm0 denotes the derivative of Vm with respect to z. In order to justify the factorisation procedure we must ensure that the differential equation (3.12) is compatible with (3.5). From the coefficients of D3 and D2 in these two equations we obtain the relations V20 /V2 = −2h 1 (α, z)

(3.13)

and V1 =

1 h 2 (α, z), 2

(3.14)

respectively, while a comparison of the coefficients of D leads to the requirement that 3 h 3 (α, z) − h 1 (α, z)h 2 (α, z) − h 02 (α, z) = 0. 2

(3.15)

If the left-hand side of Eq. (3.15) is expanded as a Laurent series about the poles at z = 0, z 1 , z 2 , z 3 , z 5 we find, by inspecting the leading-order term in each series, that the relation (3.15) can be satisfied for all α and z provided that β0 = −1, β1 = β2 = β3 = −1/2;

z 5 = 3/(4 − α 2 ), γ1 = 1/2

(3.16)

and α 6= 2. At this stage we can use Eqs. (3.16), (3.4), and (3.6) to derive unambiguous formulae for {h j (α, z) : j = 1, . . . , 4}. Finally, we must compare the coefficients of D0 in (3.5) and (3.12). Hence we obtain the further identity 1 1 V2 = h 4 (α, z) − h 1 (α, z) h 02 (α, z) − h 002 (α, z). 2 2

(3.17)

It has been checked that this formula for V2 is consistent with the earlier relation (3.13) which also gives V2 apart from a constant of integration. We see, therefore, that the factorisation procedure has been successful and the product form (3.1) is valid for all the solutions of L4 (y) = 0. It follows, therefore, that the modified Green function yG (α, z) can be written in the form " #µ ¶ ¶ 3 µ 1 Y z −1/2 z 1/2 H1 (α, z)H2 (α, z), 1− 1− yG (α, z) = z i=1 zi z5 where H1 (α, z) and H2 (α, z) are appropriate solutions of (3.8) and (3.9), respectively.

(3.18)

84

DELVES AND JOYCE

Formulae for the functions U± (α, z) in the second-order differential equations (3.8) and (3.9) are derived using Eqs. (3.11), (3.14), and (3.17). It is found that U± (α, z) =

7(2 + α 2 ) α 4 (16 − 19α 2 ) 1 3α 4 + 2+ + 12z 4z 64(1 − α 2 )(1 − α 2 z) 16(1 − α 2 z)2 +

3(2 − α)4 (2 − α)4 (8 + 32α − 23α 2 − 5α 3 ) + 128α(1 − α)[1 − (2 − α)2 z] 16[1 − (2 − α)2 z]2

−

(2 + α)4 (8 − 32α − 23α 2 + 5α 3 ) 3(2 + α)4 + 128α(1 + α)[1 − (2 + α)2 z] 16[1 − (2 + α)2 z]2

−

(4 − α 2 )2 (40 − 31α 2 ) 3(4 − α 2 )2 − 2 2 192(1 − α )[3 − (4 − α )z] 8[3 − (4 − α 2 )z]2

±

(α 2 − 1)[3 + 5(4 − α 2 )z] p . 2z[3 − (4 − α 2 )z]2 (1 − α 2 z)[1 − (2 − α)2 z][1 − (2 + α)2 z]

(3.19)

(b) Special Cases α = 1 and 2 For the particular case α = 2 the formula (3.19) reduces to U± (2, z) =

1 7 5 3 48 + + + + 2 2 4z 2z (1 − 4z) (1 − 4z) (1 − 16z)2 +

1 36 , ± √ (1 − 16z) 2z (1 − 4z)(1 − 16z)

(3.20)

while in the isotropic limit α → 1 the surd part in (3.19) vanishes and we obtain the rational function U± (1, z) ≡ U (1, z) =

4 − 52z + 375z 2 − 378z 3 + 243z 4 . 16z 2 (1 − z)2 (1 − 9z)2

(3.21)

When α = 1 we can also use the transformations H j (1, z) = z 1/2 (1 − z)1/4 (1 − 9z)1/4 Y j (z)

( j = 1, 2)

(3.22)

to express (3.1) in the alternative form y(1, z) = Y1 (z)Y2 (z).

(3.23)

By substituting (3.22) in the differential equation [D2 + U (1, z)]H j (1, z) = 0,

(3.24)

where j = 1 and 2, we find that the functions Y1 (z) and Y2 (z) are both solutions of the Heun equation [28, 29], # 3 1 z − 12 1 1 1 16 ¡ ¢ DY + ¢ Y = 0. + + ¡ D Y+ 1 z 2(z − 1) 2 z − 9 z(z − 1) z − 19 "

2

(3.25)

85

ON THE SIMPLE CUBIC GREEN FUNCTION

An investigation of the series solutions of (3.25) about z = 0 shows that the particular solutions associated with the modified Green function yG (1, z) are ¶ 1 1 3 1 1 Y1 (z) = C F , − ; , , 1, ; z , 9 12 4 4 2 µ ¶ 1 1 1 3 1 1 F , − ; , , 1, ; z , Y2 (z) = C 9 12 4 4 2 µ

(3.26) (3.27)

where C is an arbitrary constant and F(a, b; α, β, γ , δ; z) is a Heun function [28]. From Eqs. (3.23), (3.26), and (3.27) it follows that · µ ¶¸2 1 1 1 3 1 , , − ; , , 1, ; z wG(1, w) ≡ yG (1, z) = F 9 12 4 4 2

(3.28)

where z = 1/w 2 . Finally, the application of the scaling transformation formula [28, p. 119] µ

1 b z F(a, b; α, β, γ , δ; z) = F , ; α, β, γ , 1 + α + β − γ − δ; a a a

¶ (3.29)

to (3.28) gives · µ ¶¸2 3 1 3 1 . wG(1, w) ≡ yG (1, z) = F 9, − ; , , 1, ; 9z 4 4 4 2

(3.30)

This result is in agreement with the earlier work of Joyce [16, p. 588].

4. TRANSFORMATION OF THE SECOND-ORDER DIFFERENTIAL EQUATIONS FOR H1 (α, z) AND H2 (α, z) TO HYPERGEOMETRIC FORM In this section we shall show that the differential equations (3.8) and (3.9) can both be transformed by quartic transformations into the differential equation associated with the hypergeometric function 1 1 2 F1 ( 4 , 4 ; 1; x). (a) Hypergeometric Forms for wG(α, w) with α = 1 We shall first investigate some of the mathematical properties of the Green function G(α, w) for the isotropic case α = 1. From the work of Joyce [19, p. 473] it is found that G(1, w) has the following parametric representation, wG(1, w) ≡ yG (1, z) =

¸2 · (1 − 9ξ 4 ) 2 K (k) , (1 − ξ 2 )3/2 (1 − 9ξ 2 )1/2 π

(4.1)

where k2 =

(1 − 6ξ 2 − 3ξ 4 ) 1 1 − 2 2 (1 − ξ 2 )3/2 (1 − 9ξ 2 )1/2

(4.2)

86

DELVES AND JOYCE

and z ≡ 1/w2 =

4ξ 2 (1 − ξ 2 )(1 − 9ξ 2 ) . (1 − 9ξ 4 )2

(4.3)

It is possible to express the parameter ξ in terms of the variable z by using the inverse relation (1.26). Next we apply the standard result µ ¶ 2 1 1 K (k) = 2 F1 , ; 1; k 2 π 2 2

(4.4)

and the quadratic transformation formula [30, p. 111] µ 2 F1

1 1 , ; 1; k 2 2 2

¶

µ = 2 F1

¶ 1 1 , ; 1; x , 4 4

(4.5)

where x = 4k 2 (1 − k 2 ),

(4.6)

to Eq. (4.1). This procedure yields the alternative representation wG(1, w) ≡ yG (1, z) =

(1 − 9ξ 4 ) 2 (1 − ξ )3/2 (1 − 9ξ 2 )1/2

·

µ 2 F1

1 1 , ; 1; x 4 4

¶¸2 ,

(4.7)

with x =−

64ξ 6 . (1 − ξ 2 )3 (1 − 9ξ 2 )

(4.8)

The hypergeometric function 2 F1 ( 14 , 14 ; 1; x) in (4.7) is known to be a solution of the second-order differential equation [30, p. 56] 16x(1 − x)

dy F d2 y F − y F = 0. + 8(2 − 3x) 2 dx dx

(4.9)

We now eliminate the parameter ξ from Eqs. (4.3) and (4.8). Hence, we find that x = x(z) is a solution of the polynomial equation: 1 1 Q(x, z) ≡ z 4 x 4 + z(2 − z)(1 − 10z + 17z 2 )x 3 + (1 − 20z + 390z 2 − 596z 3 + 321z 4 )x 2 2 16 1 + z(2 − z)(1 − 10z + 17z 2 )x + z 4 = 0. (4.10) 2 More generally, Eq. (4.10) defines an algebraic function which has four branches {x j (z) : j = 1, 2, 3, 4}. In the neighbourhood of z = 0 we can expand these branches in the form 21 4 1413 5 22555 6 712101 7 z − z − z − z − ···, 2 16 32 128 19771 4 1246587 5 z − z − ···, x2 (z) = −16z − 152z 2 − 1247z 3 − 2 16 x1 (z) = −z 3 −

(4.11) (4.12)

ON THE SIMPLE CUBIC GREEN FUNCTION

1 21 351 253 4293 78003 2 + 2− + + z+ z + ···, z3 2z 16z 32 256 512 1 19 197 179 2 1627 3 42443 4 x 4 (z) = − + − z− z + z + z + ···, 16z 32 256 512 2048 4096 x 3 (z) = −

87 (4.13) (4.14)

where |z| ≤ 1/9. The particular branch associated with the argument x of the 2 F1 function in Eq. (4.7) is x 1 (z). We see from the symmetry of Eq. (4.10) that x3 (z) = 1/x1 (z) and x4 (z) = 1/x2 (z). It should also be noted that the set of points (x, z) which satisfy the quartic equation (4.10) forms a rational complex algebraic curve with a genus g = 0 (see [31]). From the above analysis we conclude that if α = 1 it is possible, by applying a quartic transformation x = x(z), to establish a connection between the differential equation (3.24) for {H j (1, z) : j = 1, 2} and the hypergeometric equation (4.9) for the function 2 F1 ( 14 , 14 ; 1; x). Our main aim in the remainder of this section is to show how this result can be generalized for α 6= 1. (b) Transformation Theory for α 6= 1 and the Schwarzian Derivative We begin the general analysis by applying the transformation y F = x −1/2 (1 − x)−1/4 yn

(4.15)

to the differential equation (4.9) which is associated with 2 F1 ( 14 , 14 ; 1; x). This procedure leads to the normal form (4 − 5x + 4x 2 ) d2 yn + yn = 0. dx 2 16x 2 (1 − x)2

(4.16)

We now apply a general transformation x = x(α, z) to the independent variable in Eq. (4.16), where α is taken to be a constant parameter. Hence we find that 2 x 00 (α, z) dyn d2 yn 0 2 (4 − 5x + 4x ) + [x − (α, z)] yn = 0, dz 2 x 0 (α, z) dz 16x 2 (1 − x)2

(4.17)

where the x 0 (α, z) denotes the derivative of x(α, z) with respect to z. The application of the transformation yn = [x 0 (α, z)]1/2 y

(4.18)

brings (4.17) back to the further normal form µ ¶2 · ¸ dx (4 − 5x + 4x 2 ) 1 d2 y {x, z} + + y = 0, dz 2 2 dz 16x 2 (1 − x)2

(4.19)

where {x, z} ≡

· ¸ x 000 (α, z) 3 x 00 (α, z) 2 − x 0 (α, z) 2 x 0 (α, z)

(4.20)

is the Schwarzian derivative of x(α, z) with respect to z (see [32, p. 120]). Next we consider two particular transformations x = x+ (α, z) and x = x− (α, z) for which Eqs. (4.19) are identical to the differential equations (3.8) and (3.9) for H1 (α, z) and H2 (α, z),

88

DELVES AND JOYCE

respectively. We see that x+ (α, z) and x− (α, z) must satisfy the non-linear Schwarzian differential equations 1 {x+ , z} + 2

µ

dx+ dz

¶2

(4 − 5x+ + 4x+2 ) = U+ (α, z) 16x+2 (1 − x+ )2

(4.21)

(4 − 5x− + 4x−2 ) = U− (α, z), 16x−2 (1 − x− )2

(4.22)

and 1 {x− , z} + 2

µ

dx− dz

¶2

respectively, where the algebraic functions U± (α, z) are given by (3.19). It can be shown that the lefthand sides of (4.21) and (4.22) are invariant under the transformations x+ → 1/x+ and x− → 1/x− , respectively. When α = 1 the Schwarzian differential equations (4.21) and (4.22) reduce to 1 {x, z} + 2

µ

dx dz

¶2

(4 − 5x + 4x 2 ) = U (1, z), 16x 2 (1 − x)2

(4.23)

where the rational function U (1, z) is defined in (3.21). It has been verified using the parametric equations (4.3) and (4.8) that all the solutions {x j (z) : j = 1, 2, 3, 4} of the quartic equation (4.10) satisfy Eq. (4.23). Finally, we note that differential equations of a similar type to (4.23) also play an important role in the transformation theories for elliptic integrals [33] and the 2 F1 hypergeometric function [34, 35]. (c) Algebraic Solutions of the Schwarzian Differential Equations for α 6= 1 We shall determine the transformation function x+ (α, z) by substituting a trial series solution x+ = z ζ

∞ X

cn (α)z n

(4.24)

n=0

in the Schwarzian equation (4.21). It is found that the expansion of the left-hand side of (4.21) has a leading-order term given by 1 {x+ , z} + 2

µ

dx+ dz

¶2

µ ¶ (4 − 5x+ + 4x+2 ) 1 1 , = 2 +O 2 2 4z z 16x+ (1 − x+ )

(4.25)

as z → 0. From this result we see that a series solution of the type (4.24) will only be possible if µ ¶ 1 1 , as z → 0. (4.26) U+ (α, z) = 2 + O 4z z The formula (3.19) for U+ (α, z) is consistent with (4.26). It is also evident from (4.25) that the coefficient c0 (α) and the exponent ζ are not determined by the Schwarzian differential equation. However, a comparison of (4.24) with the expansions (4.11)–(4.14) for the case α = 1 suggests that the exponent values ζ = ±1, ±3 might yield algebraic solutions of the Schwarzian equation (4.21) when α 6= 1. We now choose the exponent value ζ = 1 and determine the coefficients {cn (α) : n ≥ 1} in (4.24) by equating the coefficients of {z n : n ≥ −1} in the expansions (4.25) and (4.26) extended to higher

89

ON THE SIMPLE CUBIC GREEN FUNCTION

order. Formulae for the first few coefficients are ¤ 1£ 4(4 + 3α 2 ) − 3c0 c0 , 8 ¤ 1 £ c2 (α) = 64(84 + 160α 2 + 31α 4 ) − 384(4 + 3α 2 )c0 + 75c02 c0 , 1024 1 £ c3 (α) = 128(496 + 1780α 2 + 856α 4 + 75α 6 ) − 192(116 + 208α 2 + 49α 4 )c0 4096 ¤ + 450(4 + 3α 2 )c02 − 41c03 c0 , c1 (α) =

c4 (α) =

(4.27) (4.28)

(4.29)

£ 1 32768(12514 + 72704α 2 + 65012α 4 + 14016α 6 + 699α 8 ) 8388608 − 786432(208 + 668α 2 + 365α 4 + 42α 6 )c0 + 115200(148 + 256α 2 + 67α 4 )c02 ¤ − 167936(4 + 3α 2 )c03 + 9063c04 c0 , (4.30)

where c0 ≡ c0 (α). Results have also been derived for the higher-order coefficients {cn (α) : 5 ≤ n ≤ 20}. Next we conjecture that there exists one value of c0 (α) for which the transformation x+ (α, z) is a solution of an algebraic quartic equation of the type Q α (x, z) ≡ A4,4 (α)z 4 x 4 + x 3

4 X j=1

A3, j (α)z j + x 2

4 X

A2, j (α)z j + x

j=0

4 X

A1, j (α)z j + z 4 = 0,

j=1

(4.31) where Ai, j (α) is an unknown function of α. The structure of (4.31) has been assumed to be a simple generalisation of the quartic equation (4.10) which is valid for the isotropic case α = 1. In order to ensure that (4.31) is consistent with the symmetry properties of Eqs. (4.10) and (4.21) we also impose the conditions {A1, j (α) ≡ A3, j (α) : j = 1, 2, 3, 4}

and

A4,4 (α) ≡ 1.

(4.32)

It is possible to determine the 9 distinct functions Ai, j (α) by first using (4.24) to expand Q α (x+ , z) in powers of z. If the coefficients of {z n : n = 2, 3, . . . ,10} in the resulting series are equated to zero we obtain a set of coupled linear equations which can be solved to give the unknown functions Ai, j (α) as rational polynomials of high degree in c0 (α) and α. Then we use these formulae for Ai, j (α) to calculate the coefficients of {z n : n = 11, 12, 13} in the expansion of Q α (x+ , z). We find that these higher-order coefficients have only one common factor [c0 (α) + 16]. It appears, therefore, that x + (α, z) will be an algebraic function which satisfies a quartic equation of the type (4.31) provided that c0 (α) = −16. For this particular value of c0 (α) the complexity of the formulae for Ai, j (α) is greatly reduced and the quartic equation (4.31) becomes Q α (x, z) ≡ z 4 x 4 +

£ ¤ z 1 £ (2 − α 2 z) 1 − 2(4 + α 2 )z + (16 + α 4 )z 2 x 3 + 1 − 4(4 + α 2 )z 2α 4 16α 4

¤ + 2(176 + 16α 2 + 3α 4 )z 2 − 4(64 + 80α 2 + 4α 4 + α 6 )z 3 + (256 + 64α 4 + α 8 )z 4 x 2 £ ¤ z + 4 (2 − α 2 z) 1 − 2(4 + α 2 )z + (16 + α 4 )z 2 x + z 4 = 0. (4.33) 2α

90

DELVES AND JOYCE

When α = 1 Eq. (4.33) reduces to (4.10) and the series for x+ (1, z) with c0 (1) = −16 is in agreement with (4.12). We have rigorously verified that the transformation x + (α, z) with ζ = 1 and c0 (α) = −16 is a solution of the Schwarzian differential equation (4.21) by using the implicit equation (4.33) to evaluate the derivatives of x+ (α, z) with respect to z. It is also readily seen that the transformation 1/x+ (α, z) with ζ = 1 and c0 (α) = −16 is another solution of Eqs. (4.21) and (4.33). Originally, it was assumed that the quartic equation (4.31) had the more general form 4 X i=0

xi

4 X

Ai, j (α)z j = 0,

(4.34)

j=0

with A0,4 (α) ≡ 1, and the determination of the 24 unknown functions Ai, j (α) was carried out for several rational values of α with ζ = 1. These calculations involved very heavy algebra and large integers and it was necessary to use modular arithmetic. The advantage of the minimal assumption (4.31) is that it enables one to perform the computations for general α using ordinary exact rational arithmetic. The transformation function x− (α, z) is determined in a similar manner by substituting the trial series solution x− = z 3

∞ X

dn (α)z n

(4.35)

n=0

in the Schwarzian equation (4.22). Hence we obtain the following formulae for the first few series coefficients in (4.35), 1 (16 + 5α 2 )d0 , 2 3 d2 (α) = (256 + 192α 2 + 23α 4 )d0 , 16 1 [(8192 + 11008α 2 + 3140α 4 + 203α 6 ) − 12d0 ]d0 , d3 (α) = 32 1 d4 (α) = [(163840 + 344064α 2 + 175232α 4 + 26864α 6 + 1093α 8 ) 128

d1 (α) =

− 48(16 + 5α 2 )d0 ]d0 ,

(4.36) (4.37) (4.38)

(4.39)

where d0 ≡ d0 (α). After imposing the requirement that the series (4.35) must satisfy an algebraic equation of the type (4.31) we find that the transformation x− (α, z) is a solution of the same quartic equation (4.33), provided that d0 (α) = −α 4 . The polynomial equation (4.33) defines an algebraic function x(α, z) which has four branches. It is possible to investigate the analytic properties of x(α, z) by first calculating the discriminant Dis(Q α , x) of Q α (x, z) with respect to the variable x. We find that Dis(Q α , x) =

¤2 £ ¤2 £ ¤4 £ z4 (1 − α 2 z)2 1 − (2 − α)2 z 1 − (2 + α)2 z 1 − (4 + α 2 )z 24 256α £ ¤4 £ ¤2 × 1 + (4 − α 2 )z 1 − 2(12 + α 2 )z + (4 + α 2 )2 z 2 . (4.40)

ON THE SIMPLE CUBIC GREEN FUNCTION

91

From (4.40) we see that the zeros of the discriminant are z = 0, z 1 , z 2 , z 3 , and z 6 = (4 + α 2 )−1 ,

(4.41)

z 7 = (α 2 − 4)−1 , ³p ´2 . 8 + α2 − 2 z8 = (4 + α 2 )2 ,

(4.42)

z9 =

³p ´2 . 8 + α2 + 2 (4 + α 2 )2 ,

(4.43) (4.44)

where {z m : m = 1, 2, 3} are defined in (2.21)–(2.23). These zeros give all the singular points (or critical points) of the algebraic function x(α, z) in the finite z plane (see [36]). It can also be shown that z = ∞ is a singular point of x(α, z). In general, one finds that at least one of the branches of an algebraic function f (z) has a pole or branch-point singularity at a singular point z s . However, in exceptional circumstances, it is possible for all the branches of f (z) to be analytic at a singular point z s . We shall call a singular point of this special type a quasi-singular point. For the particular algebraic function x(α, z) we find that z = 0, {z m : m = 1, 2, 3}, and z = ∞ are normal singular points, while {z j : j = 6, 7, 8, 9} are all quasi-singular points. If we expand the square-root function in (3.19) as a Taylor series about the quasi-singular point z 7 and the introduced singularity z 5 = 3/(4 − α 2 ) then it is found that the series coefficients are all rational functions of α, while the Taylor series for the square-root function about the quasi-singular points z 8 and z 9 have coefficients which can be written in the form √ 11 + 12 8 + α 2 , where 11 and 12 are rational functions of α. When α has a general value in the interval (0, ∞), the set of points (x, z) which satisfy the equation Q α (x, z) = 0 also defines a complex elliptic curve Cα with a genus g = 1 (see [31]). For the special values α = 1 and 2 the curve Cα becomes a complex rational curve with a genus g = 0. (d) Closed-Form Solutions of the Equation Q α (x, z) = 0 The basic quartic equation (4.33) is now solved in order to obtain explicit formulae for the algebraic functions x± (α, z). We begin by introducing the transformation s=−

4x . (1 − x)2

(4.45)

If the variable x is eliminated from Eqs. (4.33) and (4.45) we obtain the quadratic form ¤4 £ ¤ 1 + (4 − α 2 )z s 2 − 32z 2 − (16 + 5α 2 )z + 4(8 + 2α 2 + α 4 )z 2 − α 2 (4 − α 2 )2 z 3 s + 256α 4 z 4 = 0.

£

(4.46) From Eqs. (4.45) and (4.46) we can derive the following formulae for the four solutions of the quartic equation (4.33), i p 1 h s+ (α, z) − 2 ± 2 1 − s+ (α, z) , s+ (α, z) i p 1 h s− (α, z) − 2 ± 2 1 − s− (α, z) , = s− (α, z)

[x+ (α, z)]±1 =

(4.47)

[x− (α, z)]±1

(4.48)

92

DELVES AND JOYCE

where s± (α, z) =

n o2 16z 2 2 12 2 12 2 12 )z] ± [1 − (2 + α) z] (1 − α z) [1 − (2 − α) z] . [1 − (4 + α [1 + (4 − α 2 )z]4 (4.49)

We shall show in Subsection 5(b), by using 2 F1 transformation theory, that the ancillary functions s± (α, z) play a very important role in the analysis of G(α, w). It should be noted that s+ (α, z 8 ) = 1 for all α ∈ (0, ∞) and s+ (α, z 7 ) = ∞, provided that α 6= 2. It is also possible to write the quartic equation (4.33) in the parametric form 16x 2 − 8η[8 − (8 + α 2 )η + 2α 2 η2 ]x + α 4 η4 = 0,

(4.50)

where the parameter η = η(α, z) satisfies the equation α 2 zη2 + [1 − (4 + α 2 )z]η + 4z = 0.

(4.51)

From the solutions of Eqs. (4.50) and (4.51) we obtain the following alternative formulae for the transformation functions x± (α, z), x± (α, z) =

o 1 n 1 1 1 η [8 − (8 + α 2 )η + 2α 2 η2 ] ± 2(1 − η) 2 (4 − α 2 η) 2 (4 − 4η + α 2 η2 ) 2 , 4

(4.52)

where η ≡ η(α, z) = −16z

´−2 ³p p 1 − (2 − α)2 z + 1 − (2 + α)2 z .

(4.53)

If the sign of one of the square roots in (4.53) is changed, then the right-hand side of Eq. (4.52) gives formulae for 1/x∓ (α, z). We shall find in Section 5 that the parametric representation (4.52) and (4.53) for x± (α, z) is more useful than the explicit formulae (4.47) and (4.48) because it enables one to determine the analytic properties of the Green function G(α, w) over a larger region of the cut w plane. (e) Transformation Formulae for H1 (α, z) and H2 (α, z) It follows from (4.15), (4.18), and the Schwarzian transformation theory developed above that any solutions H1 (α, z) and H2 (α, z) of the second-order differential equations (3.8) and (3.9), respectively, can be expressed in the form µ 1 2

1 4

1 2

1 4

H1 (α, z) = (x+ ) (1 − x+ )

µ H2 (α, z) = (x− ) (1 − x− )

dx+ dz dx− dz

¶− 12 y F(1) (x+ ),

(4.54)

y F(2) (x− ),

(4.55)

¶− 12

where {y F( j) (x) : j = 1, 2} are two (not necessarily independent) solutions of the hypergeometric equation (4.9) and the transformation functions x± = x± (α, z) are defined by the parametric equations (4.52) and (4.53). In the next section we shall use the results (4.54) and (4.55) to evaluate the Green function G(α, w) in terms of 2 F1 hypergeometric functions.

93

ON THE SIMPLE CUBIC GREEN FUNCTION

5. PRODUCT FORMULAE FOR THE GREEN FUNCTION G(α, w) The main purpose in this section is to express the Green function G(α, w) in terms of a product of two 2 F1 hypergeometric functions. A similar formula involving a product of two complete elliptic integrals of the first kind will also be given. (a) Derivation of General Product Formulae for wG(α, w) We begin by applying the transformation formulae (4.54) and (4.55) to Eq. (3.18). This procedure yields µ yG (α, z) ≡ wG(α, w) = T (α, z)(x+ x− ) 1

1 2

dx+ dz

¶− 12 µ

dx− dz

¶− 12

1

× (1 − x+ ) 4 (1 − x− ) 4 y F(1) (x+ )y F(2) (x− ),

(5.1)

where " µ ¶ 1#µ ¶1 3 1 Y z 2 z −2 T (α, z) = 1− 1− z i=1 zi z5

(5.2)

and {y F( j) (x) : j = 1, 2} are appropriate solutions of the hypergeometric differential equation (4.9). The singularity positions {z i : i = 1, 2, 3} in (5.2) are defined in (2.21)–(2.23) and z 5 is given by (3.2). The various algebraic multiplier factors in (5.1) are determined using (4.52), (4.53), and the quadratic equation (4.50). After some algebra we obtain the following result, µ 1 2

(x± ) (1 − x± )

1 4

dx± dz

¶− 12

=

¢1 1¡ (1 − η) 4 1 − 14 α 2 η 4 1

1

3 4 [T (α, z)] 2 " ×

1

1

1

1

1

1

(2 − η)(4 − α 2 η) 2 ± (1 − η) 2 (4 − 4η + α 2 η2 ) 2 (2 − η)(4 − α 2 η) 2 ∓ (1 − η) 2 (4 − 4η + α 2 η2 ) 2

# 14 , (5.3)

where the parameter η = η(α, z) is defined in (4.53). If Eq. (5.3) is applied to the formula (5.1) we find that ¶1 µ 1 1 2 2 1 yG (α, z) ≡ wG(α, w) = √ (1 − η) 2 1 − α η y F(1) (x+ )y F(2) (x− ). 4 3

(5.4)

In the neighbourhood of z = 0 the formula (5.4) must be consistent with the moment expansion (2.1). This requirement can be satisfied by taking the solutions of the hypergeometric equation (4.9) to be µ ¶ 1 1 1 4 y F(1) (x) = y F(2) (x) = 3 2 F1 , ; 1; x . (5.5) 4 4 From (5.4) and (5.5) we obtain the product formula µ wG(α, w) = (1 − η)

1 2

1 1 − α2η 4

¶ 12

µ 2 F1

1 1 , ; 1; x+ 4 4

¶

µ 2 F1

¶ 1 1 , ; 1; x− , 4 4

(5.6)

94

DELVES AND JOYCE

where x± = x± (α, z) and η = η(α, z) are defined in Eqs. (4.52) and (4.53), respectively. We have used the formula (5.6) to generate the moments {µ2n (α) : 0 ≤ n ≤ 5} in the expansion (2.1) and agreement was found with the results (2.4)–(2.9). In order to determine the region of validity in the z plane for the formula (5.6) we first construct a complex curve C0 (α) which consists of the set of points ½

·

´2 p 1³ z : x+ (α, z) ∈ 1, 2 2 + 4 + α 2 α

¸¾ .

(5.7)

The curve C0 (α) divides the z plane into two regions R0 (α) and R00 (α), where R0 (α) denotes the region which contains the origin point z = 0. It can be shown that (5.6) is valid for all points plane. When 0 < α ≤ 2 we can use (5.6) with z = 1/(u − i²)2 z ∈ R0 (α) which are in the cut z √ to calculate G R (u) and G I (u) for 4 + α 2 ≤ u < ∞, while for α > 2 it√is also possible to use (5.6) to determine G R (u) and G I (u) in the additional interval 0 < u ≤ α 2 − 4. If z ∈ R00 (α) we can modify the formula (5.6) by replacing the hypergeometric function 2 F1 ( 14 , 14 ; 1; x+ ) with the following analytic continuation formula [30, p. 108], · µ ¶ µ ¶ µ ¶ µ ¶¸ 1 1 1 3 3 3 1 1 2 1 2 3 2 , ; ; 1 − x+ ∓ 2i 0 (x+ − 1) 2 F1 , ; ; 1 − x+ , (5.8) 0 2 F1 2π 3/2 4 4 4 2 4 4 4 2 where the upper and lower signs are valid for Im(z) > 0 and Im(z) < 0, respectively. The region of validity R0 (α) for (5.6) can also be extended by applying the quadratic transformation formula (4.5) to the 2 F1 functions in (5.6). This procedure gives the alternative expression µ wG(α, w) = (1 − η)

1 2

1 1 − α2η 4

¶ 12

µ 2 F1

1 1 2 , ; 1; k+ 2 2

¶

µ 2 F1

¶ 1 1 2 , ; 1; k− , 2 2

(5.9)

where

2 k±

≡

" # ¶1 µ µ ¶1 1 1 2 2 2 1 2 2 1 = (1 − η) 2 , 1±η 1− α η − 1−η+ α η 2 4 4 ´−2 ³p p , η ≡ η(α, z) = −16z 1 − (2 − α)2 z + 1 − (2 + α)2 z

2 k± (α, z)

(5.10) (5.11)

and z = 1/w 2 . We can also use (4.4) to write (5.9) in the complete elliptic integral form µ wG(α, w) = (1 − η)

1 2

1 1 − α2η 4

¶ 12 ·

¸·

2 K (k+ ) π

¸ 2 K (k− ) . π

(5.12)

Finally, the substitution of the η formula (5.11) in Eqs. (5.10) and (5.12) leads to the explicit formula ·

¸·

2 p K (k+ ) wG(α, w) = p 2 2 π 1 − (2 − α) z + 1 − (2 + α) z 2

¸ 2 K (k− ) , π

(5.13)

95

ON THE SIMPLE CUBIC GREEN FUNCTION

where i−3 p 1 1 hp 2 2 ≡ k± (α, z) = − 1 − (2 − α)2 z + 1 − (2 + α)2 z k± 2 2 hp p √ p √ √ p √ i × 1 + (2 − α) z 1 − (2 + α) z + 1 − (2 − α) z 1 + (2 + α) z ½ hp p √ p √ 1 + (2 − α) z 1 + (2 + α) z × ±16z + 1 − α 2 z ¾ p √ p √ i2 + 1 − (2 − α) z 1 − (2 + α) z

(5.14)

and z = 1/w2 . We shall now suppose that the variable z = 1/w2 is allowed to trace out any path P which lies in a complex plane that is cut along the real axis from z = 1/(2 + α)2 to z = +∞. It is found that the 2 (α, z) map the path P into two associated paths which do not cross the straight modular functions k± line formed by the real interval [1, ∞). From this result and the properties of the hypergeometric function (4.4) it follows that the basic formulae (5.9), (5.12), and (5.13) should be valid for all values of w = u + iv which lie in the (u, v) plane, provided that a cut is made along the real axis from w = −2 − α to w = 2 + α. In order to provide a check on the results established in this section we shall use (5.9)–(5.11) with z = 1/(2 + α)2 to obtain a formula for G(α, 2 + α). After some simplifications we find that µ

1 G(α, 2 + α) = √ 2α

2 F1

1 1 2 , ; 1; k1,+ 2 2

¶

µ

2 F1

¶ 1 1 2 , ; 1; k1,− , 2 2

(5.15)

where 2 2 ≡ k1,± (α) = k1,±

√ ´ ³ √ 1 2 ∓ (2 + α)1/2 1 ± 1 + α . 2 2α

(5.16)

Next we apply the Kummer transformation formula [30, p. 105] µ 2 F1

1 1 , ; 1; k 2 2 2

¶

2 −1/2

= (1 − k )

µ 2 F1

k2 1 1 , ; 1; 2 2 2 k −1

¶ (5.17)

to (5.15). Hence we find that √ ³ ¸· ¸ ´· 2 √ 2 2 √ √ 2 1+α− 2+α K (k2,+ ) K (k2,− ) , G(α, 2 + α) = α π π

(5.18)

where k2,± ≡ k2,± (α) =

´³√ √ ´ √ 1 ³√ √ 2 1+α− 2+α 2+α± 2 . α

(5.19)

This result is in agreement with the Montroll formula (1.31). For the isotropic case α = 1 Eqs. (5.10)– (5.12) also reduce correctly to the Joyce formulae (1.21)–(1.23).

96

DELVES AND JOYCE

(b) Algebraic Simplification of the Basic Product Formula for wG(α, w) We now apply the standard transformation formula [30, p. 113] ¶ µ · ¸ 4x 1 1 1 3 , ; 1; x = (1 − x)−1/4 2 F1 , ; 1; − 2 F1 4 4 8 8 (1 − x)2

(5.20)

to both 2 F1 functions in the basic product formula (5.6). This procedure gives ¶1/2 µ 1 (1 − x+ )−1/4 (1 − x− )−1/4 wG(α, w) = (1 − η)1/2 1 − α 2 η 4 µ ¶ µ ¶ 1 3 1 3 × 2 F1 , ; 1; s+ 2 F1 , ; 1; s− , 8 8 8 8

(5.21)

where η = η(α, z) and s± = s± (α, z) are defined in Eqs. (4.53) and (4.49), respectively, and z = 1/w 2 . Next we use the quadratic equations (4.50) and (4.51) to express (5.21) in the reduced form µ ¶ µ ¶ 1 3 1 3 wG(α, w) = [1 + (4 − α 2 )z]−1/2 2 F1 , ; 1; s+ 2 F1 , ; 1; s− . (5.22) 8 8 8 8 The algebraic structure of (5.22) is considerably simpler than that of the product formulae (5.6), (5.9), and (5.13) because the 2 F1 argument functions s± (α, z) satisfy the polynomial equation (4.46) which is only of degree two in the variable s. Another important feature of (5.22) is that the values of s± (α, z) at the branch points {z j = z j (α) : j = 1, 2, 3} are all real rational functions of α when α ∈ (0, ∞). We can determine the region of validity in the z plane for the formula (5.22) by first constructing a complex curve C1 (α) which consists of the set of points {z : s+ (α, z) ∈ [1, ∞)}.

(5.23)

When 0 < α < 2 the curve C1 (α) is closed with a self-intersection point at z 7 (α), and it divides the z plane into three regions R1 (α), R01 (α), and R001 (α), where R1 (α) contains the point z = 0 and R01 (α) has a common boundary with R1 (α). For the case α ≥ 2 the curve C1 (α) is a simple curve which divides the z plane into two regions R1 (α) and R01 (α). In both cases it is found that (5.22) is valid for all points z ∈ R1 (α). When z is real and positive the formula (5.22) is valid provided that z ≤ z 8 (α), where z 8 (α) is the quasi-singular point (4.43). Because z 8 (α) < z 1 (α) for all α ∈ (0, ∞), it is evident that (5.22) has a very limited range of validity for real positive values of z. In particular, the formula (5.22) cannot be used directly to calculate the correct value of the Montroll integral G(α, 2 + α). If z ∈ R01 (α) and is sufficiently close to the boundary between R1 (α) and R01 (α) then we can modify (5.22) by replacing the hypergeometric function 2 F1 ( 18 , 38 ; 1; s+ ) with a standard analytic continuation formula of the type (5.8). In this manner, we obtain ¶ 1 3 wG(α, w) = (2π) [1 + (4 − α )z] , ; 1; s− 2 F1 8 8 µ ¶ · µ ¶ µ ¶ 3 1 3 1 1 0 , ; ; 1 − s+ × 0 2 F1 8 8 8 8 2 µ ¶¸ µ ¶ µ ¶ 7 5 7 3 5 1 (1 − s+ ) 2 2 F1 0 , ; ; 1 − s+ . + 20 8 8 8 8 2 −3/2

2

−1/2

µ

(5.24)

We can use (5.24) to determine the value of wG(α, w) in the neighbourhood of the singular point

ON THE SIMPLE CUBIC GREEN FUNCTION

97

w = 2 + α. The application of a similar linear transformation formula to the hypergeometric function 1 3 2 F1 ( 8 , 8 ; 1; s− ) enables one to express (5.24) in the alternative form ¤£ − ¤ £ + − wG(α, w) = (2π)−3 [1 + (4 − α 2 )z]−1/2 3+ 1 (α, z) + 32 (α, z) 31 (α, z) − 32 (α, z) ,

(5.25)

where µ ¶ µ ¶ µ ¶ 3 1 1 3 1 0 , ; ; 1 − s± , 2 F1 8 8 8 8 2 µ ¶ µ ¶ µ ¶ 7 5 5 7 3 1 ± 2 0 (1 − s± ) 2 F1 , ; ; 1 − s± 32 (α, z) ≡ 20 8 8 8 8 2 3± 1 (α, z) ≡ 0

(5.26) (5.27)

and s± = s± (α, z) are defined in Eq. (4.49). The result (5.25) will be used in Section 7 to derive an expansion for G(α, w) in powers of w − (2 + α). (c) Numerical Evaluation of G(α, w) in the Cut w Plane Considerable insight into the global properties of G(α, w) can be achieved by plotting threedimensional surfaces for the real and imaginary parts of G(α, u + iv) above the (u, v) plane. For example, in Fig. 1 we show the surface for the imaginary part of G(3, u + iv) which was constructed by making a direct application of Mathematica [37] to Eqs. (5.12) and (5.13). It is seen that the surface has a line of discontinuities along the cut in the (u, v) plane. The magnitude of these discontinuties is given by lim {Im[G(3, u − iv)] − Im[G(3, u + iv)]} = 2πρ(3, u),

v→0+

(5.28)

where ρ(α, u) is the density-of-states function defined in (1.15). Similar surfaces have also been obtained for many other values of α ∈ (0, ∞).

FIG. 1. The surface for Im[G(3, u + iv)] plotted above the (u, v) plane.

98

DELVES AND JOYCE

Next we use the definition (1.7) with ² = 10−12 and the basic formulae (5.12) and (5.13) to calculate the numerical values of G R (α, u) and G I (α, u) for various values of α ∈ (0, ∞) with 0 ≤ u < ∞. A graphical representation of the results is given in Fig. 2 for {α = j/2 : j = 1, . . . , 8}. The results for {α = j/2 : j = 1, 3, 5} are in agreement with the earlier work of Abe and Katsura [10].

FIG. 2. Graphs showing the variation of G R (α, u)/G I (α, 0) (dashed curves) and G I (α, u)/G I (α, 0) (continuous curves) with u r ≡ u/(2 + α) for {α = j/2 : j = 1, . . . , 8}. The values of the reduced Green functions are plotted on the vertical axis while the variable u r is plotted on the horizontal axis.

ON THE SIMPLE CUBIC GREEN FUNCTION

99

FIG. 3. Graph showing the paths traced out in the complex plane by m + (3, u) (dashed curve) and m − (3, u) (continuous curve) as u decreases from ∞ to 0.

The numerical analysis of G R (α, u) and G I (α, u) involves the evaluation of the modular functions · ¸ 1 2 α, , (5.29) m ± (α, u) ≡ lim k± ²→0+ (u − i²)2 where 0 ≤ u < ∞. In Fig. 3 we show the paths traced out in the complex plane by m ± (3, u) as u decreases from ∞ to 0. The initial and final points on these paths are given by m ± (3, ∞) = 0 and m ± (3, 0) =

´2 1 ³√ 5−1 , 12

(5.30)

respectively. Finally, in order to provide a further check on the basic formula (5.13) we have evaluated the moment integral (1.17) numerically for α = 1, 2, . . . , 5 and n = 0, 1, . . . , 5 using (1.15), (1.7), and (5.13). The results obtained were in agreement with the numerical values for µ2n (α) obtained directly from Eqs. (2.4)–(2.9). 6. SPECIAL CASES G(1, w) AND G(2, w) In this section it will be shown that wG(1, w) can be expressed in terms of the square of the complete elliptic integral K (k− ). We shall also derive various parametric representations for wG(1, w) and wG(2, w). (a) Formulae for the Isotropic Green Function wG(1, w) We begin by deriving the resultant of the polynomials Q α (x± , z) with respect to the variable z, where Q α (x, z) and x± (α, z) are defined in Eqs. (4.33) and (4.52), respectively. After inspecting the

100

DELVES AND JOYCE

various factors in this resultant it is found that x± ≡ x± (α, z) satisfy the polynomial equation X α (x+ , x− ) ≡ 2(32768 − 16384α 2 − 1536α 4 − 640α 6 + 3α 8 )x+2 x−2 + 512α 4 (8 + α 2 )x+ x− (x+ + x− )(1 + x+ x− ) − 4α 4 (128 + 96α 2 + α 4 )x+ x− (x+2 + x−2 ) − 4096α 4 x+ x− (1 + x+2 x−2 ) + α 8 (x+4 + x−4 ) = 0.

(6.1)

For general values of α ∈ (0, ∞) the set of points {(x, y) : X α (x, y) = 0} defines a complex elliptic curve Eα with a genus g = 1. However, for the special values α = 1 and 2 the curve Eα becomes a complex rational curve with a genus g = 0. 2 and k± ≡ k± (α, z) Next we make the substitutions x± = 4m ± (1 − m ± ) in (6.1), where m ± ≡ k± are the moduli which occur in the fundamental product formula (5.13). After checking the two factors in the resulting expression we find that m ± ≡ m ± (α, z) satisfy the equation £ ¤ £ Wα (m + , m − ) ≡ α 4 m 4+ − 4α 2 m − (32 + α 2 ) − 96m − + 64m 2− m 3+ + 2m − 64(2 + α 2 ) ¤ £ − (128 + 256α 2 − 3α 4 )m − + 192α 2 m 2− m 2+ − 4m − α 2 (32 + α 2 )m 2− ¤ − 32(2 + α 2 )m − + 64 m + + α 4 m 4− = 0. (6.2) When α = 1 we see that (6.2) reduces to W1 (m + , m − ) ≡ m 4+ − 4m − (33 − 96m − + 64m 2− )m 3+ + 6m − (64 − 127m − + 64m 2− )m 2+ − 4m − (33m 2− − 96m − + 64)m + + m 4− = 0.

(6.3)

This symmetric equation is just one of the well known forms for the cubic modular equation (see [13, p. 125; 38, p. 196]). Jacobi [33, p. 122] has also given the modular equation (6.3) in the more compact form (m + − m − )4 − 128m + m − (1 − m + )(1 − m − )(2 − m + − m − + 2m + m − ) = 0.

(6.4)

It follows from the cubic transformation theory of elliptic integrals (see [38]) that the ratio K (k+ )/K (k− ) must be an algebraic function of z, where k± ≡ k± (1, z) is given by (5.14). In particular, Joyce [19] has proved that K (k+ )/K (k− ) =

p p 4 − η − 1 − η,

(6.5)

where η ≡ η(1, z) is defined in Eq. (5.11). The application of (6.5) to the formula (5.12) with α = 1 yields the required result µ wG(1, w) = (1 − η)

1 2

1 1− η 4

¶ 12 ³

¸2 ´· 2 p p K (k− ) , 4−η− 1−η π

(6.6)

where η ≡ η(1, z) and k− ≡ k− (1, z). The formula (6.6) is valid for all values of w = u + iv which lie in the (u, v) plane, provided a cut is made along the real axis from w = −2 − α to w = 2 + α. It is possible to give a simple alternative derivation of the formula (6.5) by considering the functions µ 1

1

y± (α, z) ≡ (x± ) 2 (1 − x± ) 4

dx± dz

¶− 12

µ 2 F1

¶ 1 1 , ; 1; x± , 4 4

(6.7)

101

ON THE SIMPLE CUBIC GREEN FUNCTION

where x ± ≡ x± (α, z). When α = 1 it is clear from Eqs. (4.54), (4.55), and the analysis in Section 3 that y± (1, z) are both solutions of the same differential equation (3.24). It follows, therefore, that y+ (1, z) = C0 y− (1, z),

(6.8)

where C0 is a constant. By expanding both sides of Eq. (6.8) in powers of z we find that C0 = The application of (4.4) and (4.5) to (6.7) enables one to write (6.8) in the form K (k+ ) √ (x− ) 2 (1 − x− ) 4 (dx− /dz)− 2 = 3 1 1 1 , K (k− ) (x+ ) 2 (1 − x+ ) 4 (dx+ /dz)− 2 1

1

√ 3.

1

(6.9)

where x ± ≡ x± (1, z) and k± ≡ k± (1, z). We can now establish the result (6.5) by substituting the formulae (5.3) with α = 1 in the right-hand side of Eq. (6.9). Finally, we apply (4.4), (4.5), and (5.20) to the formula (6.6). This procedure gives ¶1 µ ½ µ ¶¾2 ´ p 1 2 ³p 1 3 1 1 4 − η − 1 − η (1 − x− )− 2 2 F1 , (6.10) , ; 1; s− wG(1, w) = (1 − η) 2 1 − η 4 8 8 where η ≡ η(1, z) and x− ≡ x− (1, z) are defined in (4.53) and (4.52), respectively, and s− ≡ s− (1, z) =

i2 √ 16z h 1 − 9z . (1 − 5z) − (1 − z) (1 + 3z)4

(6.11)

We can also write (6.10) in the much reduced form wG(1, w) =

¢½ ¡ √ µ ¶¾2 2 − 1 − 9z 1 3 F . , ; 1; s 2 1 − (1 + 3z) 8 8

(6.12)

This simple result can be used to calculate G(1, w) at any point in the cut w plane. For example, when w = 3 we find that G(1, 3) =

1 2

½

¶¾ µ 1 2 1 3 , ; 1; F = 0.505462019717326006 . . . . 2 1 8 8 9

(6.13)

(b) Parametric Representations for wG(1, w) When α = 1 the complex algebraic curve associated with the equation Wα (m + , m − ) = 0 has a 2 2 = k± (1, z) as rational functions genus g = 0. It follows, therefore, that we can represent m ± ≡ k± of a parameter ξ1 . In particular, we find that 2 m + ≡ k+ =−

16ξ1 , (1 + ξ1 )(1 − 3ξ1 )3

(6.14)

2 =− m − ≡ k−

16ξ13 (1 + ξ1 )3 (1 − 3ξ1 )

(6.15)

are possible representations. The dependence of the variable z on the parameter ξ1 can be found by substituting (6.14) in the polynomial equation Q 1 [4m + (1 − m + ), z] = 0, where Q 1 (x, z) is defined

102

DELVES AND JOYCE

in (4.33). This procedure gives the rational function z=

4ξ1 (1 − ξ1 )(1 + ξ1 )2 (1 + 3ξ1 )(1 − 3ξ1 )2 . ¡ ¢2 ¡ ¢2 1 + 3ξ12 1 + 6ξ1 − 3ξ12

(6.16)

(The result (6.16) can also be derived using (6.15) and Q 1 [4m − (1 − m − ), z] = 0.) The application of the formulae (6.14)–(6.16) to Eq. (5.13) with α = 1 yields the parametric representation ¢¡ ¢· ¸· ¸ 1 + 3ξ12 1 + 6ξ1 − 3ξ12 2 2 K (k K (k ) ) , wG(1, w) = + − (1 + ξ1 )2 (1 − 3ξ1 )2 π π ¡

(6.17)

2 2 ≡ k± (ξ1 ) are defined in Eqs. (6.14) and (6.15). The relation (6.5) can also be used to where k± express (6.17) in the alternative form

¡

¢¡ ¢· ¸2 1 + 3ξ12 1 + 6ξ1 − 3ξ12 2 wG(1, w) = K (k− ) . (1 + ξ1 )3 (1 − 3ξ1 ) π

(6.18)

If the Kummer transformation formula (5.17) is applied to (6.18) we find that ¢¡ ¢· ¸2 1 + 3ξ12 1 + 6ξ1 − 3ξ12 2 K (k3 ) , wG(1, w) = (1 − ξ1 )3 (1 + 3ξ1 ) π ¡

(6.19)

where k32 ≡ k32 (ξ1 ) =

16ξ13 . (1 − ξ1 )3 (1 + 3ξ1 )

(6.20)

A disadvantage of these parametric representations for wG(1, w) is that the inversion of Eq. (6.16) leads to a complicated formula for the function ξ1 = ξ1 (z). We shall now show how this difficulty can be overcome by introducing a new parameter s ξ=

ξ1 (1 − ξ1 ) . 1 + 3ξ1

(6.21)

In the first stage of the analysis the transformation formula [30, p. 113] 1 2 K (k) = 1 π (1 + k 2 ) 2

· 2 F1

4k 2 1 3 , ; 1; 4 4 (1 + k 2 )2

¸ (6.22)

is applied to (6.19). Hence we obtain #)2 ¢¡ ¢( " 1 + 3ξ12 1 + 6ξ1 − 3ξ12 64ξ13 (1 − ξ1 )3 (1 + 3ξ1 ) 1 3 ¢ 2 F1 , ; 1; ¡ . (6.23) wG(1, w) = ¡ ¢2 4 4 1 − 6ξ12 + 24ξ13 − 3ξ14 1 − 6ξ12 + 24ξ13 − 3ξ14 ¡

103

ON THE SIMPLE CUBIC GREEN FUNCTION

This result can be written in the alternative form ½

(1 − 9ξ 4 ) wG(1, w) = (1 − 6ξ 2 − 3ξ 4 )

·

2 F1

64ξ 6 1 3 , ; 1; 4 4 (1 − 6ξ 2 − 3ξ 4 )2

¸¾2 ,

(6.24)

where the parameter ξ is defined in Eq. (6.21). Next we apply the further transformation formula [30, p. 112] µ 2 F1

1 3 , ; 1; y 4 4

¶ =

(1 +

1 √

µ 1

y)2

2 F1

√ ¶ 2 y 1 1 , ; 1; √ 2 2 1+ y

(6.25)

to (6.24). This procedure gives the parametric representation · ¸ (1 − 9ξ 4 ) 2 ¡˜¢ 2 wG(1, w) = K k , (1 − ξ )3 (1 + 3ξ ) π

(6.26)

where k˜ 2 ≡ k˜ 2 (ξ ) =

16ξ 3 . (1 − ξ )3 (1 + 3ξ )

(6.27)

We can also use (6.21) to express (6.16) in the form z=

4ξ 2 (1 − ξ 2 )(1 − 9ξ 2 ) . (1 − 9ξ 4 )2

(6.28)

From Eq. (6.28) it is possible to derive the simple inverse relation ´− 12 ³ ´12 ³ √ √ 1− 1−z . ξ ≡ ξ (z) = 1 + 1 − 9z

(6.29)

The results (6.26)–(6.29) are in agreement with the earlier work of Joyce [19, 20]. A striking feature of the representations (6.19) and (6.26) is that the functions k32 (ξ1 ) and k˜ 2 (ξ ) both have the same form. It should also be noted that a further parametric representation for wG(1, w) can be obtained by making the substitution z=

ξ2 (2 − ξ2 ), 9

(6.30)

in Eqs. (6.11) and (6.12). The final result is wG(1, w) =

3 (3 − ξ2 )

½ 2 F1

·

16ξ23 (2 − ξ2 ) 1 3 , ; 1; 8 8 9(3 − ξ2 )4

¸¾2 ,

(6.31)

where ξ2 = 1 −

√

1 − 9z.

(6.32)

(c) Parametric Representations for wG(2, w) For the case α = 2 the algebraic curve associated with the equation Wα (m + , m − ) = 0 has a genus g = 1 and it is not possible to express m ± ≡ m ± (2, z) as rational functions of a parameter ξ3 .

104

DELVES AND JOYCE

However, we have seen earlier that the curve E2 associated with the equation X 2 (x+ , x− ) = 0 has a genus g = 0. Furthermore, it can also be shown using (6.1) and (6.2) that X 2 (x+ , x− ) ≡ 256 W1 (x+ , x− ).

(6.33)

It follows, therefore, from Eqs. (6.14) and (6.15) that we can formally represent x± ≡ x± (2, z) in the α = 1 parametric form x+ = −

16ξ3 , (1 + ξ3 )(1 − 3ξ3 )3

(6.34)

x− = −

16ξ33 . (1 + ξ3 )3 (1 − 3ξ3 )

(6.35)

We can determine the function z = z(ξ3 ) by substituting (6.34) in the polynomial equation Q 2 (x+ , z) = 0, where Q 2 (x, z) is defined in (4.33). In this manner we obtain z=

ξ3 (1 + ξ3 )(1 − 3ξ3 ) . (1 − ξ3 )2 (1 + 3ξ3 )2

(6.36)

We now apply the formulae (6.34)–(6.36) to Eqs. (5.6) and (4.53) with α = 2. This procedure gives the required parametric representation wG(2, w) =

(1 − ξ3 )(1 + 3ξ3 ) (1 + ξ3 )(1 − 3ξ3 )

µ 2 F1

1 1 , ; 1; x+ 4 4

¶

µ 2 F1

¶ 1 1 , ; 1; x− , 4 4

(6.37)

where x± ≡ x± (ξ3 ) are given by (6.34) and (6.35). The formula (6.37) is valid provided that z(ξ3 ) ∈ R0 (2), where R0 (α) is the region of validity in the cut z plane for (5.6). Next we use the Kummer transformation formula [30, p. 105] µ 2 F1

1 1 , ; 1; x 4 4

¶ =

1 (1 − x)1/4

µ 2 F1

x 1 3 , ; 1; 4 4 x −1

¶ (6.38)

to write (6.37) in the alternative simple form · wG(2, w) = 2 F1

16ξ3 1 3 , ; 1; 4 4 (1 − ξ3 )(1 + 3ξ3 )3

¸

· 2 F1

¸ 16ξ33 1 3 , ; 1; . 4 4 (1 − ξ3 )3 (1 + 3ξ3 )

(6.39)

We also note that the inversion of Eq. (6.36) leads to the formula ξ3 ≡ ξ3 (z) = ¡

1+

√

1 − 4z

¢¡√

4z 1 − 4z +

√

1 − 16z

¢.

(6.40)

Finally, it is possible to derive a further parametric representation for wG(2, w) by analysing the formula (5.22) with α = 2. In particular, we find that · wG(2, w) = 2 F1

64ξ4 (1 − ξ4 )3 1 3 , ; 1; 8 8 (1 + 8ξ4 )3

¸

· 2 F1

64ξ43 (1 − ξ4 ) 1 3 , ; 1; 8 8 (1 + 8ξ4 )

¸ (6.41)

ON THE SIMPLE CUBIC GREEN FUNCTION

105

and z ≡ z(ξ4 ) =

ξ4 (1 − ξ4 ) . (1 + 8ξ4 )

(6.42)

The formula (6.41) is valid provided that z(ξ4 ) ∈ R1 (2), where R1 (α) is the region of validity for (5.22).

7. BEHAVIOUR OF G(α, w) IN THE NEIGHBOURHOOD OF w = 2 + α In this section we shall derive an expansion for G(α, w) which is valid in a sufficiently small neighbourhood of the singular point w = 2 + α, where α ∈ (0, ∞). (a) Expansion for G(α, w) about w = 2 + α We begin by investigating the series solutions of the fourth-order differential equation (2.36) in the neighbourhood of the regular singular point w = 2 + α. In this manner we find that G(α, w) can be expanded in the form G(α, w) =

∞ X n=0

∞ X 1 n 1/2 A(1) (α) [w − (2 + α)] − Cn(1) (α)[w − (2 + α)]n , (7.1) [w − (2 + α)] √ n π 2α n=0

(1) where −π < arg[w − (2 + α)] < π, and the coefficients A(1) n (α) and C n (α) depend on the value (1) (1) of the parameter α. When α ∈ (0, ∞) the coefficients An (α) and Cn (α) take real values with It is also clear from (7.1) that A(1) C0(1) 0 (α) has the Montroll value G(α, 2 + α). The multiplier √ ≡¢1. ¡ (α) −1 − π 2α in the singular part of (7.1) has been obtained from the work of Maradudin et al. [6] and Iwata [25]. A striking feature of the two series in Eq. (7.1) is that they each converge in different regions of the w plane! The first series in (7.1) is convergent on the disc |w − (2 + α)| ≤ r1 (α), where

r1 (α) = 2(1 + α), = 4,

for α ≤ 1 for α > 1,

(7.2)

while the second series is convergent on the disc |w − (2 + α)| ≤ r2 (α), where r2 (α) = 2α, = 2,

for α ≤ 1 for α > 1.

(7.3)

It follows, therefore, that the complete series representation (7.1) is valid in the cut w plane provided that |w − (2 + α)| ≤ r2 (α). The application of the limiting process (1.7) to Eq. (7.1) leads to the following expansions for G R (α, u) and G I (α, u) about u = 2 + α, G R (α, u) =

∞ X

n A(1) n (α)[u − (2 + α)] ,

(7.4)

n=0

G I (α, u) =

∞ X 1 Cn(1) (α)[u − (2 + α)]n , √ [(2 + α) − u]1/2 π 2α n=0

(7.5)

where 2 + α−r2 (α) ≤ u ≤ 2 + α. When u > 2 + α it is seen that G I (α, u) ≡ 0 and we can, therefore,

106

DELVES AND JOYCE

use the expansion (7.1) with w = u to represent G R (α, u) in the interval 2 + α < u ≤ 2 + α + r2 (α). (1) (b) Formulae for the Coefficients A(1) n (α) and C n (α) (1) Formulae for {A(1) n (α) : n = 0, 1, 2, . . .} and {C n (α) : n = 0, 1, 2, . . .} in (7.1) have been derived for arbitrary values of α ∈ (0, ∞) by using computer algebra methods to expand the right-hand side of Eq. (5.25) in powers of w − (2 + α). In particular, we find that

A(1) 0 (α) ≡ G(α, 2 + α) = A(1) 1 (α) =

¤ £ 1 (2 + α)2 221 (ϑ1 ) − 16(1 + α)222 (ϑ1 ) , 5/2 + α)

16π 3 (2

£ 1 −36(8 + α)(2 + α)6 221 (ϑ1 ) − 81α 3 (1 + α)(2 + α)2 223 (ϑ1 ) 9216π 3 (2 + α)15/2 + 216α 2 (2 + α)4 21 (ϑ1 )23 (ϑ1 ) + 576(2 + α)4 (8 + 9α − 7α 2 + 4α 3 )222 (ϑ1 ) ¤ + 19600α 3 (1 + α)2 224 (ϑ1 ) − 13440α 2 (1 − α 2 ) (2 + α)2 22 (ϑ1 )24 (ϑ1 ) ,

A(1) 2 (α) =

(7.6)

442368π 3 (1

(7.7)

£ 1 108(4 + α)(5 + α)(4 + 3α)(2 + α)6 221 (ϑ1 ) 17/2 + α)(2 + α)

+ 81α 2 (1 + α)(2 + α)2 (16 + 32α + 41α 2 + 13α 3 ) 223 (ϑ1 ) − 1296α 2 (2 + α)4 (7 + 8α + 2α 2 )21 (ϑ1 )23 (ϑ1 ) − 576(1 + α)(2 + α)4 (240 + 288α − 179α 2 + 25α 3 + 52α 4 )222 (ϑ1 ) − 19600α 2 (1 + α)2 (16 + 32α + 41α 2 + 13α 3 )224 (ϑ1 )

¤ + 13440α 2 (1 − α 2 )(2 + α)2 (26 + 42α + 13α 2 )22 (ϑ1 )24 (ϑ1 ) ,

(7.8)

where 21 (ϑ) ≡ 22 (ϑ) ≡ 23 (ϑ) ≡ 24 (ϑ) ≡

¶ µ ¶ µ ¶ µ 3 1 1 3 1 0 , ; ;ϑ , 0 2 F1 8 8 8 8 2 µ ¶ µ ¶ ¶ µ 5 7 5 7 3 0 F 0 , ; ; ϑ , 2 1 8 8 8 8 2 µ ¶ µ ¶ ¶ µ 1 3 9 11 3 0 0 , ; ;ϑ , 2 F1 8 8 8 8 2 µ ¶ µ ¶ ¶ µ 5 7 13 15 5 0 0 , ; ;ϑ , 2 F1 8 8 8 8 2

(7.9) (7.10) (7.11) (7.12)

and ϑ1 ≡ ϑ1 (α) =

4(1 + α) . (2 + α)2

(7.13)

For the coefficient C0(1) (α) we obtain the expression C0(1) (α) =

√ £ α2 2 12(2 + α)2 21 (ϑ1 )22 (ϑ1 ) − 9(1 + α)22 (ϑ1 )23 (ϑ1 ) 48π 2 (2 + α)4 ¤ + 35(1 + α)21 (ϑ1 )24 (ϑ1 ) .

(7.14)

ON THE SIMPLE CUBIC GREEN FUNCTION

107

The application of the known result C0(1) (α) = 1 to (7.14) yields the hypergeometric relation √ 96π 2 2 , 4821 (ϑ)22 (ϑ) − 9ϑ22 (ϑ)23 (ϑ) + 35ϑ21 (ϑ)24 (ϑ) = (1 − ϑ)

(7.15)

where ϑ ∈ (0, 1). This identity can be used to simplify the formulae for all the higher-order coefficients {Cn(1) (α) : n = 1, 2, . . .}. In this manner, we find that 1 (1 + 2α), 12α ¢ 1 ¡ C2(1) (α) = 9 + 4α + 20α 2 , 2 480α ¡ ¢ 1 C3(1) (α) = − 75 + 18α + 20α 2 + 168α 3 . 3 13440α C1(1) (α) = −

(7.16) (7.17) (7.18)

It is possible to express all the functions {2 j (ϑ) : j = 1, 2, 3, 4} in terms of complete elliptic integrals of the first and second kinds by using the formulae √ ¤ £ √ 21 (ϑ) = 2 π(1 + k 2 )1/2 K 0 (k) + 2K (k) , √ √ ¤ π(1 + k 2 )5/2 £ 0 K (k) − 2K (k) , 22 (ϑ) = 2 4 (1 − 6k + k ) √ √ £ 2 0 2 π(1 + k 2 )9/2 23 (ϑ) = − 2 2k K (k) − 2(1 − k 2 ) K (k) 2 2 2 4 3k (1 − k ) (1 − 6k + k ) √ ¤ − (1 + k 2 ) E 0 (k) + 2(1 + k 2 ) E(k) , √ £ 2 3 π(1 + k 2 )13/2 2k (9 − 22k 2 + 9k 4 ) K 0 (k) 24 (ϑ) = − 2 2 2 2 4 3 35k (1 − k ) (1 − 6k + k ) √ + 2(1 − k 2 ) (1 − 22k 2 + 17k 4 ) K (k) − (1 + k 2 ) (1 − 6k 2 + k 4 ) E 0 (k) √ ¤ − 2(1 + k 2 ) (1 − 6k 2 + k 4 ) E(k) ,

(7.19) (7.20)

(7.21)

(7.22)

where K 0 (k) and E 0 (k) are complementary elliptic integrals of the first and second kinds, respectively, ¡ ϑ ≡ ϑ(k) =

1 − 6k 2 + k 4 ¡ ¢4 1 + k2

¢2 (7.23)

√ and k ∈ (0, 1) with k 6= 2 − 1. Next we define the modulus k in (7.19)–(7.23) to be the α dependent function k ≡ k(α) =

´³√ √ ´ √ 1 ³√ √ 2 1+α− 2+α 2+α− 2 , α

(7.24)

where α ∈ (0, ∞). (It should be noted that in the notation of the Montroll formula (5.18) we have

108

DELVES AND JOYCE

k(α) ≡ k2,− (α).) It can be shown that the inverse of the function (7.24) is given by ¡ ¢ 8k 1 − k 2 α ≡ α(k) = ¡ ¢2 , 1 − 2k − k 2

(7.25)

¢ ¡ √ where k ∈ 0, 2 − 1 . If Eq. (7.25) is substituted in the right-hand side of the formula (7.13) we find that ϑ1 (α) 7→ ϑ(k), where ϑ(k) is defined in (7.23). Under these circumstances, we can use (7.19)–(7.22) and (7.25) to express the formulae (7.6)–(7.8) in the alternative complete elliptic integral form ¢ 1 ¡ 1 − 2k − k 2 K 0 (k)K (k), A(1) 0 (α) ≡ G(α, 2 + α) = 2 π ¡ ¢ ¢¡ ¢ £ ¡ 1 − 2k − k 2 (1) ¡ ¢¡ ¢ k 1 − k 2 1 − k + k 3 + k 4 K 0 (k)K (k) A1 (α) = 4π 2 k 1 − k 4 1 + 2k − k 2 ¡ ¢¡ ¢ ¢ ¡ − 1 − k 2 1 + 3k 2 + 2k 3 E 0 (k)K (k) − 2k 1 − k + k 3 + k 4 E(k)K 0 (k) ¤ ¢¡ ¢ ¡ + 1 + k 2 1 + 2k − k 2 E 0 (k)E(k) , ¡ ¢ £ ¡ ¢¡ 1 − 2k − k 2 (1) 2 2 A2 (α) = ¡ ¢2 ¡ ¢3 ¡ ¢3 − k 1 − k 29 − 18k 2 2 2 2 2 192π k 1 − k 1+k 1 + 2k − k

(7.26)

(7.27)

+ 3k 2 + 160k 3 + 45k 4 − 18k 5 − 261k 6 + 240k 7 + 351k 8 − 278k 9 − 111k 10 ¡ ¢ ¢¡ + 48k 11 − 41k 12 − 6k 13 − 15k 14 K 0 (k)K (k) + 1 − k 2 1 + 6k + 55k 2 + 176k 3 −39k 4 + 326k 5 − 201k 6 + 240k 7 + 411k 8 − 622k 9 − 195k 10 + 32k 11 + 11k 12 ¢ ¡ − 30k 13 − 43k 14 E 0 (k)K (k) + 4k 2 11 − 6k + 11k 2 + 52k 3 + 39k 4 − 74k 5 − 153k 6 ¢ + 120k 7 + 153k 8 − 74k 9 − 39k 10 + 52k 11 − 11k 12 − 6k 13 − 11k 14 E(k)K 0 (k) ¢¡ ¢¡ ¡ − 1 + k 2 1 + 2k − k 2 1 + 4k + 90k 2 − 36k 3 + 15k 4 + 152k 5 − 148k 6 ¤ ¢ − 152k 7 + 15k 8 + 36k 9 + 90k 10 − 4k 11 + k 12 E 0 (k)E(k) , (7.28) where the modulus k = k(α) is defined in (7.24) and α ∈ (0, ∞). The simple formula (7.26) for A(1) 0 (α) can also be derived from the Montroll result (5.18) by applying the transformation K (k2,+ ) =

1 (1 + k)K 0 (k), 2

(7.29)

where k2,+ = k2,+ (α) and k = k(α) are defined in Eqs. (5.19) and (7.24), respectively. If the expansion (7.1) is substituted in the differential equation (2.36) it is found that, in general, (1) the coefficients A(1) n (α) and C n (α) both satisfy nine-term recurrence relations. (For the special cases (1) α = 1 and 8 the recurrence relations associated with A(1) n (α) and C n (α) can be reduced to four and eight-term relations, respectively.) These relations and the initial conditions C0(1) (α) ≡ 1 and (7.26)– (7.28) are sufficient to enable one to successively generate explicit formulae for all the higher-order coefficients {Cn(1) (α) : n = 1, 2, . . .} and {A(1) n (α) : n = 3, 4, . . .}.

ON THE SIMPLE CUBIC GREEN FUNCTION

109

(c) Singular Value Theory 2 We shall now investigate the possibility of expressing A(1) 0 (α) ≡ G(α, 2 + α) in terms of [K (k)] . In the first stage of the analysis we note that the function k(α), defined in (7.24), is monotonic in the interval α ∈ (0, ∞). It is evident, therefore, that the equation

k(α) = k[N ],

(7.30)

where k[N ] is the singular value of order N , will have a unique solution α ≡ α[N ], provided √ that 0 < k[N√ ] < 2 − 1. From the known values of k[N ] it is found [14] that the inequality 0 < k[N ] < 2 − 1 is satisfied for all integer values of N ≥ 3. We also see from (7.25) that the value of α[N ] is given by α[N ] =

8k[N ](1 − k 2 [N ]) , (1 − 2k[N ] − k 2 [N ])2

(7.31)

where N ≥ 3. The application of Eqs. (1.20) and (7.30) to the formula (7.26) leads to the required result √

¾ ½ ¢ 2 ¢ 2 ¡ N¡ 2 1 − 2k[N ] − k [N ] K k[N ] , G(α[N ], 2 + α[N ]) = 4 π

(7.32)

where N = 3, 4, . . . . For the particular case N = 6 we find that the substitution of the singular value ³ √ ´ √ ´ ³√ 3− 2 k[6] = 2 − 3

(7.33)

in (7.31) gives α[6] = 1. Under these circumstances (7.32) reduces to the Watson formula (1.18) for G(1, 3). The singular values k[10], k[18], k[22], and k[58] also lead to simple rational values for α[N ]. For example, when N = 58 it is found that k[58] =

´³ √ √ ´ 1³ √ 99 2 − 140 26 29 − 99 2 , 2

(7.34)

with α[58] = 1/4900. Hence we obtain the new formula µ G

1 9801 , 4900 4900

¶

√ ¾ ½ ¢ 2 ¢ 2 ¡ 58 ¡ = 1 − 2k[58] − k 2 [58] K k[58] . 4 π

(7.35)

For the special cases {α = α[N ] : N = 3, 4, . . .} it can also be shown using generalised singular value theory [13, p. 152] that all the coefficients {A(1) n (α) : n = 1, 2, . . .} can be expressed in terms of just one complete elliptic integral K (k[N ]). (d) Generating Function for the Coefficient Cn(1) (α) Finally, we note that the method used by Maradudin et al. [6] to analyse the behaviour of G(α, w) near w = 2 + α enables one to establish the identity · 2 F0

µ

1 1 z , ; 2 2 2

¶¸2

µ 2 F0

1 1 z , ; 2 2 2α

¶ =

∞ µ ¶ X 3 n=0

2

n

Cn(1) (α)(−z)n ,

(7.36)

110

DELVES AND JOYCE

where 2 F0 denotes a generalized hypergeometric series. Although the 2 F0 series are divergent for all z 6= 0, we can still consider (7.36) to be a formal generating function for the coefficients {Cn(1) (α) : n = 0, 1, 2, . . .}. If the expansion formula [30, p. 187] ·

µ 2 F0

1 1 z , ; 2 2 2

¶¸2 =

∞ X

¡ 1 ¢2 2 n

n!

n=0



1 , 2

−n,

 3 F2 

− n,

1 2

1 2



1 ; 2

 −1

− n;

µ ¶n z 2

(7.37)

is substituted in (7.36) we obtain the closed-form expression

α n Cn(1) (α) =

(−1)n

¡1¢ 2 n

n µ ¶ X n

n!2n (2n + 1) m=0 m

¡ 1 ¢2 ¡1 2



1 , 2

−m,

 m ¢2 α 3 F2 

1 ; 2

2 m

−n

1 2

m

− m,

1 2

  −1 ,

(7.38)

− m;

where n = 0, 1, 2, . . .. It does not appear to be possible to sum the well-poised 3 F2 hypergeometric series in (7.38) in terms of gamma functions. However, we can use a quadratic transformation formula [39, p. 267] to express the terminating 3 F2 series in the alternative form  3 F2

1 , 2

−m,

  1 2

− m,

1 2

1 ; 2





  m −1 = 2 3 F2 

− m;

− m2 ,

−m, 1 2

− m,

1 2

1 2

−

m ; 2

  1 .

(7.39)

− m;

8. BEHAVIOUR OF G(α, w) IN THE NEIGHBOURHOOD OF w = α Our main aim in this section is to derive an expansion for G(α, w) which is valid in a sufficiently small neighbourhood of the singular point w = α, where α ∈ (0, ∞). (a) Expansion for G(α, w) about w = α From the differential equation (2.36) and the work of Iwata [25] we find that G(α, w) can be expanded about the regular singular point w = α in the form r ∞ ∞ X X i 2 (2) n G(α, w) = (w − α)1/2 Dn (α)(w − α) − Cn(2) (α)(w − α)n , (8.1) π α n=0 n=0 where −π < arg(w − α) < 0 and C0(2) (α) ≡ 1, provided that α 6= 1. When α = 1 the expansion (8.1) breaks down because the branch-point singularities at w = α and w = 2 − α are confluent in the limit α → 1. It can be shown that (8.1) gives a convergent representation for G(α, w) which is valid in the lower half of the cut w plane provided that |w − α| ≤ r3 (α), where r3 (α) = 2α,

for α ≤

1 2

= 2(α − 1),

1 <α<1 2 for 1 < α ≤ 2

= 2,

for α > 2.

= 2(1 − α),

for

(8.2)

ON THE SIMPLE CUBIC GREEN FUNCTION

111

In the following analysis we shall not deal with the special case α = 1 because the behaviour of G(1, w) in the neighbourhood of w = 1 has already been investigated in detail by Joyce [16, p. 594]. The coefficients {Dn(2) (α) : n = 0, 1, 2, . . .} are complex-valued functions of the parameter α, while the coefficients {Cn(2) (α) : n = 0, 1, 2, . . .} are real functions of α. If we make the substitution w = α − i² in (8.1) and take the limit ² → 0+, then it is clear from (1.7) that D0(2) (α) ≡ G − (α, α) = G R (α, α) + iG I (α, α).

(8.3)

(b) Formulae for the Coefficients Dn(2) (α) and Cn(2) (α) In order to obtain explicit expressions for the coefficients in the expansion (8.1) we first use the methods described in Subsection 5(b) to establish the following analytic continuation formula, ³ i √ ´ + √ ´ £ ¡ ¢ ¤−1/2 h³ 1 + i 2 3+ uG − (α, u) = (2π)−3 1 + 4 − α 2 z 1 (α, z) − 1 − i 2 32 (α, z) ¤ £ − (8.4) × 3− 1 (α, z) − 32 (α, z) , where 0 < α < 1, z = 1/u 2 , and {3±j (α, z) : j = 1, 2} are defined in (5.26) and (5.27), respectively. The result (8.4) is valid in the immediate neighbourhood of the singular point w = α, provided that u ≥ α. We can now use computer algebra methods to expand the right-hand side of (8.4) in powers of u − α. In this manner, it is found that (2) (2) Dn(2) (α) ≡ A(2) n,+ (α) + An,− (α) + iAn,3 (α)

(n = 0, 1, 2, . . .),

(8.5)

where ¡ ¤ ¢ 1 £ 2 (8.6) 2 (ϑ2 ) + 4 1 − α 2 222 (ϑ2 ) , 16π 3 1 ¢1/2 1 ¡ = − 3 1 − α2 21 (ϑ2 )22 (ϑ2 ), (8.7) 4π √ ¡ ¤ ¢ 2 £ 2 = (8.8) 21 (ϑ2 ) − 4 1 − α 2 222 (ϑ2 ) , 3 16π £ ¡ ¢ α = 144221 (ϑ2 ) − 81α 2 1 − α 2 223 (ϑ2 ) − 216α 2 21 (ϑ2 )23 (ϑ2 ) 18432π 3 ¢ ¡ ¢2 ¡ + 576 1 − 25α 2 222 (ϑ2 ) − 4900α 2 1 − α 2 224 (ϑ2 ) ¤ ¡ ¢ (8.9) − 16800α 2 1 − α 2 22 (ϑ2 )24 (ϑ2 ) , ¡ ¢ 1/2 ¢ ¡ ¢ £ ¡ α 1 − α2 ¡ ¢ − 48 1 − 5α 2 21 (ϑ2 )22 (ϑ2 ) + 180α 2 1 − α 2 22 (ϑ2 )23 (ϑ2 ) = 3 2 1536π 1 − α ¤ ¡ ¢2 ¡ ¢ + 105α 2 1 − α 2 23 (ϑ2 )24 (ϑ2 ) + 140α 2 1 − α 2 21 (ϑ2 )24 (ϑ2 ) , (8.10) √ ¡ ¢ α 2 £ = 144221 (ϑ2 ) − 81α 2 1 − α 2 223 (ϑ2 ) − 216α 2 21 (ϑ2 )23 (ϑ2 ) 18432π 3 ¡ ¢ ¡ ¢2 − 576 1 − 25α 2 222 (ϑ2 ) + 4900α 2 1 − α 2 224 (ϑ2 ) ¤ ¡ ¢ (8.11) + 16800α 2 1 − α 2 22 (ϑ2 )24 (ϑ2 )

A(2) 0,+ (α) = A(2) 0,− (α) A(2) 0,3 (α) A(2) 1,+ (α)

A(2) 1,− (α)

A(2) 1,3 (α)

112

DELVES AND JOYCE

and £ ¡ ¢ ¡ ¢ 1 ¡ ¢ 432 4 − α 2 221 (ϑ2 ) + 81α 2 4 − 5α 2 + α 4 223 (ϑ2 ) 442368π 3 1 − α 2 ¡ ¢ − 1296α 2 21 (ϑ2 )23 (ϑ2 ) + 576 12 + α 2 − 13α 4 222 (ϑ2 ) ¤ ¡ ¢2 ¡ ¢ ¡ ¢ + 4900α 2 1 − α 2 4 − α 2 224 (ϑ2 ) + 6720α 2 5 − 7α 2 + 2α 4 22 (ϑ2 )24 (ϑ2 ) , (8.12) ¡ ¢1/2 £ ¡ ¢ 1 − α2 (2) 2 4 A2,− (α) = ¡ ¢2 −48 4 − 13α + α 21 (ϑ2 )22 (ϑ2 ) 3 2 12288π 1 − α ¡ ¢ ¡ ¢2 ¡ ¢ − 24α 2 5 − 7α 2 + 2α 4 22 (ϑ2 )23 (ϑ2 ) − 35α 2 1 − α 2 4 − α 2 23 (ϑ2 )24 (ϑ2 ) ¤ ¡ ¢ (8.13) + 280α 2 1 − α 2 21 (ϑ2 )24 (ϑ2 ) , √ £ ¡ ¢ ¡ ¢ 2 ¡ ¢ 432 4 − α 2 221 (ϑ2 ) + 81α 2 4 − 5α 2 + α 4 223 (ϑ2 ) A(2) 2,3 (α) = 3 2 442368π 1 − α ¡ ¢ − 1296α 2 21 (ϑ2 )23 (ϑ2 ) − 576 12 + α 2 − 13α 4 222 (ϑ2 ) ¤ ¡ ¢2 ¡ ¢ ¡ ¢ − 4900α 2 1 − α 2 4 − α 2 224 (ϑ2 ) − 6720α 2 5 − 7α 2 + 2α 4 22 (ϑ2 )24 (ϑ2 ) . (8.14) A(2) 2,+ (α) =

The functions {2 j (ϑ) : j = 1, 2, 3, 4} in the formulae (8.6)–(8.14) are given by Eqs. (7.9)–(7.12) and ϑ2 ≡ ϑ2 (α) = 1 − α 2 .

(8.15)

In the above analysis we have assumed that α ∈ (0, 1). However, the final results (8.6)–(8.14) are also found to be valid for α ∈ (1, ∞), provided that the radical (1 − α 2 )1/2 in Eqs. (8.7), (2) (8.10), and (8.13) is replaced by +i(α 2 − 1)1/2 . It should be noted that A(2) n,+ (α) and An,3 (α) are (2) real functions of α in the intervals (0, 1) and (1, ∞) while An,− (α) has a real value for α ∈ (0, 1) and a purely imaginary value for α ∈ (1, ∞). These properties are important because they enable one to obtain exact formulae for the real and imaginary parts of the complex-valued coefficient Dn(2) (α). The formulae for {Cn(2) (α) : n = 0, 1, 2, . . .} can all be simplified by using the identity (7.15). Hence, we find that 1 , 12α ¢ 1 ¡ C2(2) (α) = 9 + 16α 2 , 2 480α ¡ ¢ 1 C3(2) (α) = − 75 + 16α 2 , 3 13440α

C1(2) (α) = −

(8.16) (8.17) (8.18)

where α ∈ (0, ∞), with α 6= 1. In order to transform the formulae (8.6)–(8.14) to complete elliptic integral form we first introduce the α dependent modulus k ≡ k(α) =

q ´³ ´ p p 1 ³√ 2 1 − 1 − α2 − α 1 + 1 − α2 , 2 α

(8.19)

113

ON THE SIMPLE CUBIC GREEN FUNCTION

where α ∈ (0, ∞). The inverse of the function (8.19) is given by ¢ ¡ 4k 1 − k 2 α ≡ α(k) = ¡ ¢2 . 1 + k2

(8.20)

If Eq. (8.20) is substituted in the right-hand side of formula (8.15) we find that ϑ2 (α) 7→ ϑ(k), where ϑ(k) is defined in (7.23). Under these circumstances, we can use (7.19)–(7.22) and (8.20) to express (8.6)–(8.14) in the alternative form ¢© ª 1 ¡ 1 + k 2 2K 2 (k) ± [K 0 (k)]2 , 2 2π ¢ 2 ¡ A(2) 1 + k 2 K 0 (k)K (k), 0,3 (α) = 2 π

A(2) 0,± (α) =

A(2) 1,± (α) =

(8.22)

© (1 + k 2 ) −2(1 − k 2 )3 K 2 (k) + 4(1 − k 2 )(1 − 3k 2 )E(k)K (k) 4π 2 k(1 − k 2 )(1 − 6k 2 + k 4 ) − 2(1 − 6k 2 + k 4 )E 2 (k) ± k 2 (1 + k 2 )2 [K 0 (k)]2 ∓ 4k 2 (1 + k 2 )E 0 (k)K 0 (k) ª ∓ (1 − 6k 2 + k 4 )[E 0 (k)]2 ,

A(2) 1,3 (α) =

(8.21)

(8.23)

£ 2 (1 + k 2 ) −k (1 − k 4 )K 0 (k)K (k) − (1 − k 2 )(1 − 3k 2 )E 0 (k)K (k) 2 2 4 − k )(1 − 6k + k ) ¤ + 2k 2 (1 + k 2 )E(k)K 0 (k) + (1 − 6k 2 + k 4 )E 0 (k)E(k) , (8.24) π 2 k(1

and A(2) 2,± (α) =

© (1 + k 2 ) 2(1 − k 2 )2 (1 − 15k 2 − 39k 4 − 63k 6 − 213k 8 96π 2 k 2 (1 − k 2 )2 (1 − 6k 2 + k 4 )3 + 75k 10 − 5k 12 + 3k 14 )K 2 (k) − 4(1 − k 4 )(1 − 12k 2 − 3k 4 − 96k 6 + 27k 8 + 12k 10 + 7k 12 )E(k)K (k) + 2(1 + k 2 )2 (1 − 6k 2 + k 4 )(1 + 14k 4 + k 8 )E 2 (k) ± k 2 (3 + 4k 2 + 108k 4 − 324k 6 + 162k 8 − 324k 10 + 108k 12 + 4k 14 + 3k 16 )[K 0 (k)]2 ∓ 16k 2 (1 + k 2 )(1 + 3k 4 − 24k 6 + 3k 8 + k 12 )E 0 (k)K 0 (k) ª ± (1 + k 2 )2 (1 − 6k 2 + k 4 )(1 + 14k 4 + k 8 )[E 0 (k)]2 ,

A(2) 2,3 (α) =

24π 2 k 2 (1

(8.25)

£ 2 (1 + k 2 ) −k (1 − k 2 )(1 − 3k 2 )(5 + 24k 2 − 3k 4 + 72k 6 2 2 2 4 3 − k ) (1 − 6k + k )

− 33k 8 − k 12 )K 0 (k)K (k) + (1 − k 4 )(1 − 12k 2 − 3k 4 − 96k 6 + 27k 8 + 12k 10 +7k 12 )E 0 (k)K (k) + 8k 2 (1 + k 2 )(1 + 3k 4 − 24k 6 + 3k 8 + k 12 )E(k)K 0 (k) ¤ − (1 + k 2 )2 (1 − 6k 2 + k 4 )(1 + 14k 4 + k 8 )E 0 (k)E(k) ,

(8.26)

(2) (2) where k = k(α) is defined in (8.19). Recurrence relations for A(2) n,± (α), An,3 (α), and C n (α), can be derived by substituting (8.1) in the differential equation (2.36). These relations and the initial conditions (8.21)–(8.26) and C0(2) (α) ≡ 1 enable one to generate formulae for the higher-order (2) (2) coefficients {A(2) n,± (α), An,3 (α) : n = 3, 4, . . .} and {C n (α) : n = 1, 2, . . .}.

114

DELVES AND JOYCE

Finally, we note that the application of (8.21), (8.22), and (8.5) to Eq. (8.3) yields the formula G − (α, α) =

2 (1 + k 2 )K (k)[K (k) + iK 0 (k)], π2

(8.27)

where α ∈ (0, ∞). From this result it is clear that 2 (1 + k 2 )K 2 (k), π2 2 G I (α, α) = 2 (1 + k 2 )K 0 (k)K (k), π

G R (α, α) =

(8.28) (8.29)

provided that 0 < α ≤ 1. For the more difficult case 1 < α < ∞ we find that © ª 1 (1 + k 2 ) 2K 2 (k) + [K 0 (k)]2 , 2π 2 © ª 1 G I (α, α) = (1 + k 2 ) 4K 0 (k)K (k) − 2iK 2 (k) + i[K 0 (k)]2 , 2π 2

G R (α, α) =

(8.30) (8.31)

where k = k(α) is given by Eq. (8.19). (c) Singular Value Theory Our aim now is to investigate the possibility of expressing G − (α, α) in terms of [K (k)]2 . We begin the analysis by noting that the√function k(α), defined in (8.19), is real and monotonic in the interval 0 < α ≤ 1, with 0 < k(α) ≤ 2 − 1. It follows, therefore, that the equation k(α) = k[N ],

(8.32)

where k[N ] is the singular value of order N , will have a unique real solution α = α[N ], provided that N = 2, 3, . . . . We see from (8.20) that the value of α[N ] is given by α[N ] =

4k[N ] (1 − k 2 [N ]) , (1 + k 2 [N ])2

(8.33)

where N = 2, 3, . . . . The application of Eqs. (1.20) and (8.32) to the formula (8.27) gives the required result G − (α[N ], α[N ]) =

√ ´¡ ¢ ¡ ¢ 2 ³ N 1 + k 2 [N ] K 2 k[N ] . 1 + i 2 π

For the particular case N = 2 we find that k[2] = G − (1, 1) =

(8.34)

√ 2 − 1, α[2] = 1, and

´i2 √ ´h ³√ √ ´³ 4 ³ 2 2 − 2 K 2 − 1 . 1 + i π2

This formula is in agreement with the work of Katsura et al. [40] and Joyce [16].

(8.35)

115

ON THE SIMPLE CUBIC GREEN FUNCTION

It should also be noted that the singular values {k[N ] : N = 6, 10, 18, 22, 58} are all associated with rational values of α[N ]. For example, when N = 10 it is found that G

−

µ

1 1 , 9 9

¶ =

√ ´¡ ¢ 2 ³ 10 1 + k 2 [10] K 2 (k[10]), 1 + i 2 π

(8.36)

where k[10] =

³√ ´³√ ´2 10 − 3 2−1 .

(8.37)

When {α = α[N ] : N = 3, 4, . . .} we can use generalised singular value theory in order to express the higher-order coefficients {Dn(2) (α) : n = 1, 2, . . .} in terms of just one complete elliptic integral K (k[N ]). (d) Generating Function for the Coefficient Cn(2) (α) The behaviour of G(α, w) in the neighbourhood of the branch-point singularity w = α was analysed by Iwata [25] using Laplace integral representations. By extending the method of Iwata it is possible to establish the identity µ 2 F0

1 1 z , ; 2 2 2

¶

µ

2 F0

z 1 1 , ;− 2 2 2

¶

µ

2 F0

1 1 z , ; 2 2 2α

¶ =

∞ µ ¶ X 3

2

n=0

Cn(2) (α)(−z)n ,

(8.38)

n

where 2 F0 denotes a divergent generalised hypergeometric series. We can consider (8.38) to be a formal generating function for the coefficients {Cn(2) (α) : n = 0, 1, . . .}. If the expansion formula [30, p. 187] µ 2 F0

1 1 z , ; 2 2 2

¶

µ 2 F0

z 1 1 , ;− 2 2 2

¶ =

∞ X

¡ 1 ¢2 2 n

n!

n=0



1 , 2

−n,

 3 F2  1 2

− n,

1 2

1 ; 2

− n;

  1

µ ¶n z 2

(8.39)

is substituted in (8.38) we obtain the closed-form expression

α n Cn(2) (α) =

(−1)n

¡1¢

n ³ X n´



¡ 1 ¢2

¢ α ¡ n!2n (2n + 1) m=0 m 1 − n 2 2 m 2 n

2 m

m

3 F2

1 , 2

−m,

  1 2

− m,

1 2

1 ; 2

  1 ,

(8.40)

− m;

where n = 0, 1, 2, . . . . Apart from a change in the sign of the argument in the 3 F2 function this result is the same as the formula (7.38) for the coefficient Cn(1) (α). Next we note that the left-hand side of (8.39) is an even function of z. It follows, therefore, that  3 F2

1 , 2

−n,

  1 2

− n,

1 2

1 ; 2

  1 = 0,

(8.41)

− n;

whenever n is a positive odd integer. If n = 2k, where k = 0, 1, 2, . . . , then we can use the theory

116

DELVES AND JOYCE

of well-poised hypergeometric series [41, 42] to prove the identity 

1 , 2

−2k,

 3 F2  1 2

− 2k,



¡ 1 ¢4  2 k 1 = ¡ 1 ¢2 ¡ 3 ¢2 .

1 ; 2

− 2k;

1 2

(8.42)

4 k 4 k

Finally, we apply the results (8.41) and (8.42) to Eq. (8.40). This procedure leads to the surprising simple formula

α n Cn(2) (α) =

(−1)

n



¡1¢

n!2n (2n

2 n

+ 1)

5 F4

− n2 ,

  1 4

−

− n2 + 12 ,

n , 2

1 4

−

n , 2

1 , 2

3 4

−

1 , 2 n , 2

3 4

−

1 ; 2

  α2 ,

(8.43)

n ; 2

where n = 0, 1, 2, . . . . (e) Expansions for G R (α, u) and G I (α, u) about u = α We can now use (1.7), (8.1), and the results in Subsection 8(b) to obtain expansions for G R (α, u) and G I (α, u) which are valid when u ' α. For the case 0 < α < 1 it is found that G R (α, u) =

∞ X £

¤ (2) n A(2) n,+ (α) + An,− (α) (u − α) ,

n=0

G I (α, u) =

∞ X

A(2) n,3 (α)(u

n=0

1 − α) − π

r

n

∞ X 2 Cn(2) (α)(u − α)n , (u − α)1/2 α n=0

when u ≥ α; and G R (α, u) =

∞ X £

A(2) n,+ (α)

+

¤ A(2) n,− (α) (u

n=0

G I (α, u) =

1 − α) − π

r

n

∞ X

(8.44)

(8.45)

∞ X 2 Cn(2) (α)(u − α)n , (8.46) (α − u)1/2 α n=0

n A(2) n,3 (α)(u − α) ,

(8.47)

n=0

when u ≤ α. For the case 1 < α < ∞ we find that G R (α, u) =

∞ X

n A(2) n,+ (α)(u − α) ,

n=0

G I (α, u) =

∞ X £

−iA(2) n,− (α)

+

¤ A(2) n,3 (α) (u

n=0

1 − α) − π n

when u ≥ α; and G R (α, u) =

∞ X

A(2) n,+ (α)(u

n=0

G I (α, u) =

∞ X £

1 − α) − π n

r

r

∞ X 2 (u − α)1/2 Cn(2) (α)(u − α)n , α n=0

∞ X 2 Cn(2) (α)(u − α)n , (α − u)1/2 α n=0

¤ (2) n −iA(2) n,− (α) + An,3 (α) (u − α) ,

n=0

(8.48)

(8.49)

(8.50)

(8.51)

ON THE SIMPLE CUBIC GREEN FUNCTION

117

when u ≤ α. It should be noted that the coefficient −iA(2) n,− (α) in (8.49) and (8.51) has a real value for all α ∈ (1, ∞). 9. BEHAVIOUR OF G(α, w) IN THE NEIGHBOURHOOD OF w = |2 − α| We shall now investigate the behaviour of G(α, w) in the neighbourhood of the singular point w = |2 − α|, where α ∈ (0, ∞). (a) Expansions for G(α, w) about w = |2 − α| When 0 < α < 2 we can use the differential equation (2.36) and the work of Iwata [25] to expand G(α, w) about the regular singular point w = 2 − α in the form G(α, w) =

∞ X n=0

∞ X i Dn(3) (α)[w − (2 − α)]n − √ [w − (2 − α)]1/2 Cn(3) (α)[w − (2 − α)]n , (9.1) π 2α n=0

where −π < arg [w − (2 − α)] < 0 and C0(3) (α) ≡ 1, provided that α 6= 1. For the special case α = 1 the expansion (9.1) breaks down because the branch-point singularities at w = 2 − α and w = α are confluent in the limit α → 1. The expansion (9.1) gives a convergent representation for G(α, w) which is valid in the lower half of the cut w plane provided that |w − (2 − α)| ≤ r4 (α), where r4 (α) = 2α,

for α ≤

= 2(1 − α), = 2(α − 1), = 2(2 − α),

1 2

1 <α<1 2 3 for 1 < α ≤ 2 3 for < α < 2. 2 for

(9.2)

We shall not deal with the case α = 1 in this section because the behaviour of G(1, w) in the neighbourhood of w = 1 has already been investigated in detail by Joyce [16, p. 594]. When α > 2 we can write the expansion for G(α, w) about the regular singular point w = α − 2 in the form G(α, w) =

∞ X n=0

∞ X 1 Dn(4) (α)[w − (α − 2)]n + √ [w − (α − 2)]1/2 Cn(4) (α)[w − (α − 2)]n , (9.3) π 2α n=0

where −π < arg [w − (α − 2)] < 0 and C0(4) (α) ≡ 1. This expansion provides one with a convergent representation for G(α, w) which is valid in the lower half of the cut w plane provided that |w − (α − 2)| ≤ r5 (α), where r5 (α) = 2(α − 2), = 2,

for 2 < α ≤ 3

for α > 3.

(9.4)

For the case α = 2 the expansions (9.1) and (9.3) break down because the branch-point singularities at w = α − 2 and w = 2 − α are confluent in the limit α → 2.

118

DELVES AND JOYCE

The coefficients {Dn(3) (α), Dn(4) (α) : n = 0, 1, 2, . . .} are complex-valued functions of the parameter α, while the coefficients {Cn(3) (α), Cn(4) (α) : n = 0, 1, 2, . . .} are real functions of α. If we make the substitution w = 2 − α − i² in (9.1) and take the limit ² → 0+, then it is clear from (1.7) that D0(3) (α) ≡ G − (α, 2 − α) = G R (α, 2 − α) + iG I (α, 2 − α),

(9.5)

where 0 < α < 2. In a similar manner, we find from (9.3) that D0(4) (α) ≡ G − (α, α − 2) = G R (α, α − 2) + iG I (α, α − 2),

(9.6)

where α > 2. (b) Formulae for the Coefficients Dn(3) (α) and Cn(3) (α) We can derive explicit expressions for the coefficients in the expansion (9.1) by making use of the analytic continuation formula uG − (α, u) = (2π)−3 [1 + (4 − α 2 )z]−1/2

³ i √ ´ − √ ´ (α, z) + 1 − i 2 3 (α, z) 1 + i 2 3− 1 2

h³

+ × [3+ 1 (α, z) − 32 (α, z)],

(9.7)

where 0 < α < 1, z = 1/u 2 , and {3±j (α, z) : j = 1, 2} are defined in (5.26) and (5.27), respectively. This result is valid in the immediate neighbourhood of the singular point w = 2 − α, provided that u ≥ 2 − α. The application of computer algebra methods enables one to expand the right-hand side of (9.7) in powers of u − (2 − α). In this manner we obtain the formulae £ ¤ (3) (3) Dn(3) (α) ≡ i A(3) n,+ (α) + An,− (α) + An,3 (α)

(n = 0, 1, 2, . . .),

(9.8)

where √ A(3) 0,+ (α)

=

16π 3 (2

¤ £ 2 (2 − α)2 221 (ϑ3 ) + 16(1 − α)222 (ϑ3 ) , 5/2 − α)

(9.9)

i(α − 1)1/2 21 (ϑ3 )22 (ϑ3 ), (9.10) √ π 3 2(2 − α)3/2 ¤ £ 1 A(3) (9.11) (2 − α)2 221 (ϑ3 ) − 16(1 − α)222 (ϑ3 ) , 0,3 (α) = 16π 3 (2 − α)5/2 √ £ 2 (3) −36(8 − α)(2 − α)6 221 (ϑ3 ) + 81α 3 (1 − α)(2 − α)2 223 (ϑ3 ) A1,+ (α) = 9216π 3 (2 − α)15/2

A(3) 0,− (α) =

+ 216α 2 (2 − α)4 21 (ϑ3 )23 (ϑ3 ) − 576(2 − α)4 (8 − 9α − 7α 2 − 4α 3 )222 (ϑ3 ) ¤ (9.12) +19600α 3 (1 − α)2 224 (ϑ3 ) + 13440α 2 (1 − α 2 )(2 − α)2 22 (ϑ3 )24 (ϑ3 ) , √ £ i 2 12(2 − α)4 (8 − 9α − 3α 2 )21 (ϑ3 )22 (ϑ3 ) A(3) 1,− (α) = 3 1/2 13/2 384π (α − 1) (2 − α) − 36α 2 (1 − α 2 )(2 − α)2 22 (ϑ3 )23 (ϑ3 ) − 105α 3 (1 − α)2 23 (ϑ3 )24 (ϑ3 ) ¤ − 140α 2 (1 − α)(2 − α)2 21 (ϑ3 )24 (ϑ3 ) ,

(9.13)

119

ON THE SIMPLE CUBIC GREEN FUNCTION

A(3) 1,3 (α) =

£ 1 −36(8 − α)(2 − α)6 221 (ϑ3 ) + 81α 3 (1 − α)(2 − α)2 223 (ϑ3 ) 9 216π 3 (2 − α)15/2 + 216α 2 (2 − α)4 21 (ϑ3 )23 (ϑ3 ) + 576(2 − α)4 (8 − 9α − 7α 2 − 4α 3 )222 (ϑ3 ) ¤ (9.14) − 19600α 3 (1 − α)2 224 (ϑ3 ) − 13440α 2 (1 − α 2 )(2 − α)2 22 (ϑ3 )24 (ϑ3 ) ,

and A(3) 2,+ (α)

√ £ 2 = 108(4 − α)(5 − α)(4 − 3α)(2 − α)6 221 (ϑ3 ) 3 17/2 442368π (1 − α)(2 − α) + 81α 2 (1 − α)(2 − α)2 (16 − 32α + 41α 2 − 13α 3 )223 (ϑ3 ) − 1296α 2 (2 − α)4 (7 − 8α + 2α 2 )21 (ϑ3 )23 (ϑ3 ) + 576(1 − α)(2 − α)4 (240 − 288α − 179α 2 − 25α 3 + 52α 4 )222 (ϑ3 ) + 19600α 2 (1 − α)2 (16 − 32α + 41α 2 − 13α 3 )224 (ϑ3 )

¤ − 13440α 2 (1 − α 2 )(2 − α)2 (26 − 42α + 13α 2 )22 (ϑ3 )24 (ϑ3 ) , √ i 2 (3) A2,− (α) = 6144π 3 (α − 1)3/2 (2 − α)15/2 £ × 12(2 − α)4 (80 − 176α + 71α 2 + 30α 3 − 13α 4 )21 (ϑ3 )22 (ϑ3 )

(9.15)

− 12α 2 (1 − α 2 )(2 − α)2 (26 − 42α + 13α 2 )22 (ϑ3 )23 (ϑ3 ) + 35α 2 (1 − α)2 (16 − 32α + 41α 2 − 13α 3 )23 (ϑ3 )24 (ϑ3 ) ¤ − 280α 2 (1 − α)(2 − α)2 (7 − 8α + 2α 2 )21 (ϑ3 )24 (ϑ3 ) , A(3) 2,3 (α) =

(9.16)

£ 1 108(4 − α)(5 − α)(4 − 3α)(2 − α)6 221 (ϑ3 ) 442368π 3 (1 − α)(2 − α)17/2 + 81α 2 (1 − α)(2 − α)2 (16 − 32α + 41α 2 − 13α 3 )223 (ϑ3 ) − 1296α 2 (2 − α)4 (7 − 8α + 2α 2 )21 (ϑ3 )23 (ϑ3 ) − 576(1 − α)(2 − α)4 (240 − 288α − 179α 2 − 25α 3 + 52α 4 )222 (ϑ3 ) − 19600α 2 (1 − α)2 (16 − 32α + 41α 2 − 13α 3 )224 (ϑ3 )

¤ + 13440α 2 (1 − α 2 )(2 − α)2 (26 − 42α + 13α 2 )22 (ϑ3 )24 (ϑ3 ) .

(9.17)

The functions {2 j (ϑ) : j = 1, 2, 3, 4} in the formulae (9.9)–(9.17) are given by Eqs. (7.9)–(7.12) and ϑ3 ≡ ϑ3 (α) = 4(1 − α)/(2 − α)2 .

(9.18)

In the above analysis it has been assumed that α ∈ (0, 1). However, it is found that the final results (9.9)–(9.17) are also valid for α ∈ (1, 2). When α ∈ (0, 1) the radical (α − 1)1/2 in (9.10), (9.13), (3) and (9.16) has the principal value +i(1 − α)1/2 . It should be pointed out that A(3) n,+ (α) and An,3 (α) (3) are real functions of α in the intervals (0, 1) and (1, 2), while An,− (α) has a real value for α ∈ (0, 1) and a purely imaginary value for α ∈ (1, 2). These properties are important because they enable one to derive exact formulae for the real and imaginary parts of the complex-valued coefficient Dn(3) (α).

120

DELVES AND JOYCE

The formulae for the coefficients {Cn(3) (α) : n = 0, 1, 2, . . .} are found to be 1 (1 − 2α), 12α 1 C2(3) (α) = (9 − 4α + 20α 2 ), 480α 2 1 C3(3) (α) = (75 − 18α + 20α 2 − 168α 3 ), 13440α 3 C1(3) (α) =

(9.19) (9.20) (9.21)

with C0(3) (α) ≡ 1, where α ∈ (0, 2) and α 6= 1. A comparison of these results with (7.16)–(7.18) suggests that Cn(3) (α) = Cn(1) (−α)

(n = 0, 1, 2, . . .),

(9.22)

where Cn(1) (α) is defined in Eq. (7.1). The validity of this relation has been confirmed by analysing the structure of the higher-order coefficients {Cn(3) (α) : n = 4, 5, . . .}. A closed-form expression for Cn(3) (α) can now be derived by applying Eq. (9.22) to (7.38). We also note the further relations ϑ3 (α) = ϑ1 (−α), A(3) n,3 (α)

=

A(1) n (−α),

(9.23) (9.24)

where n = 0, 1, 2, . . . . In order to transform the formulae (9.9)–(9.17) to complete elliptic integral form we first introduce the α dependent modulus ´³√ ´ √ √ i ³√ √ 2 α − 1− α − 2 2+i α − 2 , (9.25) k ≡ k(α) = α √ √ where α ∈ (0, 2).√The radical α − 1 in (9.25) takes √ its principal value +i 1 − α for α ∈ (0, 1), 2). When α ∈ (0, 1) it is while the radical α − 2 has the principal value +i 2 − α for all α ∈ (0, √ found that the modulus k(α) has a real value which lies in the interval (0, 2 − 1). The inverse of the function (9.25) is given by α ≡ α(k) =

8k(1 − k 2 ) . (1 + 2k − k 2 )2

(9.26)

If the formula (9.26) is substituted in the right-hand side of Eq. (9.18) we see that ϑ3 (α) 7→ ϑ(k), where ϑ(k) is defined in (7.23). Under these circumstances, we can use (7.19)–(7.22) and (9.26) to express (9.9)–(9.17) in the alternative form © ª 1 (1 + 2k − k 2 ) 2K 2 (k) ± [K 0 (k)]2 , (9.27) 2 2π 1 (1 + 2k − k 2 )K 0 (k)K (k), (9.28) A(3) 0,3 (α) = π2 © (1 + 2k − k 2 ) 2(1 − k 2 )(1 + k + 4k 2 − 2k 3 − k 4 + k 5 )K 2 (k) A(3) 1,± (α) = 2 4 2 8π k(1 − k )(1 − 2k − k )

A(3) 0,± (α) =

− 4(1 − k 2 )(1 + 3k 2 − 2k 3 )E(k)K (k) + 2(1 + k 2 )(1 − 2k − k 2 )E 2 (k) ∓ k(1 + k 2 )(1 + k − k 3 + k 4 )[K 0 (k)]2 ± 4k(1 + k − k 3 + k 4 )E 0 (k)K 0 (k) ª (9.29) ± (1 + k 2 )(1 − 2k − k 2 )[E 0 (k)]2 ,

121

ON THE SIMPLE CUBIC GREEN FUNCTION

£ (1 + 2k − k 2 ) k(1 − k 2 )(1 + k − k 3 + k 4 )K 0 (k)K (k) 2 4 2 4π k(1 − k )(1 − 2k − k )

A(3) 1,3 (α) =

+ (1 − k 2 )(1 + 3k 2 − 2k 3 )E 0 (k)K (k) − 2k(1 + k − k 3 + k 4 )E(k)K 0 (k) ¤ − (1 + k 2 )(1 − 2k − k 2 )E 0 (k)E(k) ,

(9.30)

and A(3) 2,± (α) =

(1 + 2k − k 2 ) 384π 2 k 2 (1 − k 2 )2 (1 + k 2 )3 (1 − 2k − k 2 )3 © × 2(1 − k 2 )2 (1 − 6k + 27k 2 − 200k 3 − 15k 4 − 366k 5 − 261k 6 − 624k 7 + 411k 8 + 238k 9 − 135k 10 − 72k 11 − 13k 12 + 6k 13 − 15k 14 )K 2 (k) − 4(1 − k 2 )(1 − 6k + 55k 2 − 176k 3 − 39k 4 − 326k 5 − 201k 6 − 240k 7 + 411k 8 + 622k 9 − 195k 10 − 32k 11 + 11k 12 + 30k 13 − 43k 14 )E(k)K (k) + 2(1 + k 2 )(1 − 2k − k 2 )(1 − 4k + 90k 2 + 36k 3 + 15k 4 − 152k 5 − 148k 6 + 152k 7 + 15k 8 − 36k 9 + 90k 10 + 4k 11 + k 12 )E 2 (k) ± k 2 (1 + k 2 )(15 + 6k + 55k 2 − 36k 3 + 59k 4 + 154k 5 − 365k 6 − 376k 7 + 365k 8 + 154k 9 − 59k 10 − 36k 11 − 55k 12 + 6k 13 − 15k 14 )[K 0 (k)]2 ∓ 8k 2 (11 + 6k + 11k 2 − 52k 3 + 39k 4 + 74k 5 − 153k 6 − 120k 7 + 153k 8 + 74k 9 − 39k 10 − 52k 11 − 11k 12 + 6k 13 − 11k 14 )E 0 (k)K 0 (k) ± (1 + k 2 )(1 − 2k − k 2 )(1 − 4k + 90k 2 + 36k 3 + 15k 4 − 152k 5 ª − 148k 6 + 152k 7 + 15k 8 − 36k 9 + 90k 10 + 4k 11 + k 12 )[E 0 (k)]2 ,

A(3) 2,3 (α) =

(9.31)

(1 + 2k − k 2 ) 192π 2 k 2 (1 − k 2 )2 (1 + k 2 )3 (1 − 2k − k 2 )3 £ × − k 2 (1 − k 2 )(29 + 18k + 3k 2 − 160k 3 + 45k 4 + 18k 5 − 261k 6 − 240k 7 + 351k 8 + 278k 9 − 111k 10 − 48k 11 − 41k 12 + 6k 13 − 15k 14 )K 0 (k)K (k) + (1 − k 2 )(1 − 6k + 55k 2 − 176k 3 − 39k 4 − 326k 5 − 201k 6 − 240k 7 + 411k 8 + 622k 9 − 195k 10 − 32k 11 + 11k 12 + 30k 13 − 43k 14 )E 0 (k)K (k) + 4k 2 (11 + 6k + 11k 2 − 52k 3 + 39k 4 + 74k 5 − 153k 6 − 120k 7 + 153k 8 + 74k 9 − 39k 10 − 52k 11 − 11k 12 + 6k 13 − 11k 14 )E(k)K 0 (k) − (1 + k 2 )(1 − 2k − k 2 )(1 − 4k + 90k 2 + 36k 3 + 15k 4 − 152k 5 − 148k 6 ¤ + 152k 7 + 15k 8 − 36k 9 + 90k 10 + 4k 11 + k 12 )E 0 (k)E(k) , (9.32)

where the modulus k = k(α) is defined in (9.25). It should be noted that the formulae for {A(3) n,3 (α) : n = 0, 1, 2} can also be obtained by making the formal substitution k 7→ −k in the complete elliptic integral expressions for {A(1) n (α) : n = 0, 1, 2}, respectively. (3) (3) Recurrence relations for A(3) n,± (α), An,3 (α), and C n (α) can be derived by substituting (9.1) in the differential equation (2.36). These relations and the initial conditions (9.27)–(9.32) and C0(3) (α) ≡ 1

122

DELVES AND JOYCE

(3) enable one to generate exact formulae for the higher-order coefficients {A(3) n,± (α), An,3 (α) : n = 3, 4, . . .} and {Cn(3) (α) : n = 1, 2, . . .}. It follows from (9.26) and the discussion in Subsection 7(c) that if

α = α[N ] ≡

8k[N ](1 − k 2 [N ]) , (1 + 2k[N ] − k 2 [N ])2

(9.33)

where k[N ] is the singular value of order N and N ≥ 3, then it is possible to express all the coefficients {Dn(3) (α) : n = 0, 1, 2, . . .} in terms of just one complete elliptic integral K (k[N ]). Finally, we apply (9.27), (9.28), and (9.8) to Eq. (9.5). This procedure yields the formula G − (α, 2 − α) =

1 (1 + 2k − k 2 )K (k)[K 0 (k) + 2iK (k)], π2

(9.34)

where 0 < α ≤ 2. From this result it is clear that 1 (1 + 2k − k 2 )K 0 (k)K (k), π2 2 G I (α, 2 − α) = 2 (1 + 2k − k 2 )K 2 (k), π

G R (α, 2 − α) =

(9.35) (9.36)

provided that 0 < α ≤ 1. For the more difficult case 1 < α ≤ 2 we find that © ª 1 (1 + 2k − k 2 ) 2K 0 (k)K (k) + 2iK 2 (k) − i[K 0 (k)]2 , 2 2π © ª 1 (1 + 2k − k 2 ) [K 0 (k)]2 + 2K 2 (k) , G I (α, 2 − α) = 2 2π

G R (α, 2 − α) =

(9.37) (9.38)

where k = k(α) is given by Eq. (9.25). (c) Expansions for G R (α, u) and G I (α, u) about u = 2 − α for α ∈ (0, 2) We can now use (1.7), (9.1), and the results in Subsection 9(b) to obtain expansions for G R (α, u) and G I (α, u) which are valid when u ' 2 − α and α ∈ (0, 2) with α 6= 1. For the case 0 < α < 1 it is found that

G R (α, u) =

∞ X

n A(3) n,3 (α)[u − (2 − α)] ,

(9.39)

n=0

G I (α, u) =

∞ X £

¤ (3) n A(3) n,+ (α) + An,− (α) [u − (2 − α)]

n=0 ∞ X 1 − √ [u − (2 − α)]1/2 Cn(3) (α)[u − (2 − α)]n , π 2α n=0

(9.40)

ON THE SIMPLE CUBIC GREEN FUNCTION

123

when u ≥ 2 − α; and G R (α, u) =

∞ X n=0

∞ X 1 n [(2 − α) − u]1/2 A(3) Cn(3) (α)[u − (2 − α)]n , (9.41) n,3 (α)[u − (2 − α)] − √ π 2α n=0

G I (α, u) =

∞ X £

¤ (3) n A(3) n,+ (α) + An,− (α) [u − (2 − α)] ,

(9.42)

¤ (3) n iA(3) n,− (α) + An,3 (α) [u − (2 − α)] ,

(9.43)

n=0

when u ≤ 2 − α. For the case 1 < α < 2 we find that ∞ X £

G R (α, u) =

n=0

G I (α, u) =

∞ X

n A(3) n,+ (α)[u − (2 − α)] −

n=0

∞ X 1 Cn(3) (α)[u − (2 − α)]n , √ [u − (2 − α)]1/2 π 2α n=0

(9.44) when u ≥ 2 − α; and G R (α, u) =

∞ X £

¤ (3) n iA(3) n,− (α) + An,3 (α) [u − (2 − α)]

n=0 ∞ X 1 − √ [(2 − α) − u]1/2 Cn(3) (α)[u − (2 − α)]n , π 2α n=0

G I (α, u) =

∞ X

n A(3) n,+ (α)[u − (2 − α)] ,

(9.45)

(9.46)

n=0

when u ≤ 2 − α. It should be noted that the coefficient i A(3) n,− (α) in (9.43) and (9.45) has a real value for all α ∈ (1, 2). (d) Formulae for the Coefficients Dn(4) (α) and Cn(4) (α) In order to derive expressions for the coefficients in the expansion (9.3) we first consider the analytic continuation formula ¤−1/2 + £ − − [32 (α, z) − 3+ uG − (α, u) = (2π)−3 1 + (4 − α 2 )z 1 (α, z)][31 (α, z) + 32 (α, z)],

(9.47)

where α ∈ (2, ∞), z = 1/u 2 , and {3±j (α, z) : j = 1, 2} are defined in (5.26) and (5.27), respectively. This result is valid in the immediate neighbourhood of the singular point w = α − 2, provided that u ≥ α − 2. We can now use computer algebra methods to expand the right-hand side of (9.47) in powers of u − (α − 2). In this manner we obtain the formulae D0(4) (α) =

¤ (2 − α)2 221 (ϑ3 ) − 16(1 − α)222 (ϑ3 ) ,

£

i 16π 3 (α

−

2)5/2

(9.48)

124

DELVES AND JOYCE

D1(4) (α) =

£ i −36(8 − α)(2 − α)6 221 (ϑ3 ) + 81α 3 (1 − α)(2 − α)2 223 (ϑ3 ) 9216π 3 (α − 2)15/2 + 216α 2 (2 − α)4 21 (ϑ3 )23 (ϑ3 ) + 576(2 − α)4 (8 − 9α − 7α 2 − 4α 3 )222 (ϑ3 ) ¤ (9.49) − 19600α 3 (1 − α)2 224 (ϑ3 ) − 13440α 2 (1 − α 2 )(2 − α)2 22 (ϑ3 )24 (ϑ3 ) ,

D2(4) (α) =

442368π 3 (1

£ i 108(4 − α)(5 − α)(4 − 3α)(2 − α)6 221 (ϑ3 ) 17/2 − α)(α − 2)

+ 81α 2 (1 − α)(2 − α)2 (16 − 32α + 41α 2 − 13α 3 )223 (ϑ3 ) − 1296α 2 (2 − α)4 (7 − 8α + 2α 2 )21 (ϑ3 )23 (ϑ3 ) − 576(1 − α)(2 − α)4 (240 − 288α − 179α 2 − 25α 3 + 52α 4 )222 (ϑ3 ) − 19600α 2 (1 − α)2 (16 − 32α + 41α 2 − 13α 3 )224 (ϑ3 )

¤ + 13440α 2 (1 − α 2 )(2 − α)2 (26 − 42α + 13α 2 )22 (ϑ3 )24 (ϑ3 ) ,

(9.50)

where α ∈ (2, ∞). The functions {2 j (ϑ) : j = 1, 2, 3, 4} in the formulae (9.48)–(9.50) are given by Eqs. (7.9)–(7.12) and ϑ3 ≡ ϑ3 (α) is defined in (9.18). The formulae for the remaining coefficients {Cn(4) (α) : n = 0, 1, 2, . . .} are found to be 1 (1 − 2α), 12α 1 C2(4) (α) = (9 − 4α + 20α 2 ), 480α 2 1 C3(4) (α) = − (75 − 18α + 20α 2 − 168α 3 ), 13440α 3 C1(4) (α) = −

(9.51) (9.52) (9.53)

with C0(4) (α) ≡ 1, where α ∈ (2, ∞). A comparison of these results with (7.16)–(7.18) suggests that Cn(4) (α) = (−1)n Cn(1) (−α)

(n = 0, 1, 2, . . .),

(9.54)

where Cn(1) (α) is defined in Eq. (7.1). The validity of this relation has been confirmed by analysing the structure of the higher-order coefficients {Cn(4) (α) : n = 4, 5, . . .}. The transformation of the formulae (9.48)–(9.50) to complete elliptic integral form can be carried out in the usual manner by using the modulus (9.25) with α ∈ (2, ∞). We find that the final expressions for {(−1)n Dn(4) (α), n = 0, 1, 2} in terms of k ≡ k(α) are formally given by the right-hand side of Eqs. (9.28), (9.30), and (9.32), respectively. In particular, we have D0(4) (α) ≡ G − (α, α − 2) =

1 (1 + 2k − k 2 )K 0 (k)K (k), π2

(9.55)

where k ≡ k(α) is defined in (9.25) and α ∈ (2, ∞). From this result and (9.6) we see that G R (α, α − 2) = 0, G I (α, α − 2) = − where α ∈ (2, ∞).

(9.56) i (1 + 2k − k 2 )K 0 (k)K (k), π2

(9.57)

125

ON THE SIMPLE CUBIC GREEN FUNCTION

(e) Expansions for G R (α, u) and G I (α, u) about u = α − 2 for α ∈ (2, ∞) We can now use (1.7), (9.3), and the results in Subsection 9(d) to obtain expansions for G R (α, u) and G I (α, u) which are valid when u ' α − 2 and α ∈ (2, ∞). For the case u ≥ α − 2 these expansions are given by G R (α, u) =

∞ X 1 Cn(4) (α)[u − (α − 2)]n , √ [u − (α − 2)]1/2 π 2α n=0

G I (α, u) = −

∞ X

iDn(4) (α)[u − (α − 2)]n .

(9.58)

(9.59)

n=0

When u ≤ α − 2 we find that G R (α, u) ≡ 0, G I (α, u) = −

∞ X

iDn(4) (α)[u − (α − 2)]n −

n=0

(9.60) ∞ X

1 Cn(4) (α)[u − (α − 2)]n . √ [(α − 2) − u]1/2 π 2α n=0 (9.61)

If we apply the relation (9.54) to the expansion (9.58) and take the limit α tends to +∞, then it is seen that G R (α, u) ∼

∞ X 1 (−1)n Cn(1) (−∞)[u − (α − 2)]n , √ [u − (α − 2)]1/2 π 2α n=0

(9.62)

as α → +∞, where u ≥ α − 2 and Cn(1) (±∞) ≡ lim Cn(1) (±α).

(9.63)

α→+∞

It readily follows from (7.38) that ¡ ¢

Cn(1) (±∞) =

(−1)n 12 n 3 F2 n! 2n (2n + 1)



1 , 2

−n,

  1 2

− n,

1 2

1 ; 2

  −1  ,

(9.64)

− n;

where n = 0, 1, 2, . . . . We also find from the expansion (7.5) that G I (α, u) ∼

∞ X 1 (−1)n Cn(1) (+∞)[(α + 2) − u]n , √ [(α + 2) − u]1/2 π 2α n=0

(9.65)

as α → +∞, where α ≤ u ≤ 2 + α. A comparison of this result with the expansion (9.62) yields the symmetry relation G R (α, α − 2 + u 0 ) ∼ G I (α, α + 2 − u 0 ),

(9.66)

as α tends to +∞, where 0 < u 0 ≤ 2. An inspection of the graphs in Fig. 2 shows that this asymptotic symmetry property holds to a good approximation for quite small values of α ≥ 2.

126

DELVES AND JOYCE

10. BEHAVIOUR OF G(α, w) IN THE NEIGHBOURHOOD OF w = 0 In this section we shall derive expansions for G(α, w) which are valid in a sufficiently small neighbourhood of the point w = 0. (a) Expansion for G(α, w) about w = 0 for α ∈ (0, 2) We can investigate the behaviour of G(α, w) in the neighbourhood of w = 0 by using computer algebra methods to expand the formula (5.13) about the point z = ∞. When 0 < α < 2 it is found that G(α, w) = i

∞ X

2n A(5) n (α)w +

n=0

∞ X 2w C (5) (α)w2n , π α(4 − α 2 )1/2 n=0 n

(10.1)

where −π < arg(w) < 0 and C0(5) (α) ≡ 1. It can be shown that the expansion (10.1) provides one with a convergent representation for G(α, w) which is valid in the lower half of the cut w plane provided that |w| ≤ r6 (α), where r6 (α) = α, = 2 − α,

for α ≤ 1 for 1 < α < 2.

(10.2)

The expansion (10.1) breaks down when α = 2 because the branch-point singularities w = 2 − α and w = −2 + α are confluent with the origin w = 0 in the limit α → 2. The first few coefficients {Cn(5) (α), A(5) n (α) : n = 0, 1, 2} in the expansion (10.1) are given by the formulae C0(5) (α) ≡ 1,

(10.3) 2 (4 − 2α 2 + α 4 ), − α 2 )2

(10.4)

8 (36 − 40α 2 + 18α 4 + α 8 ) − α 2 )4

(10.5)

C1(5) (α) =

3α 2 (4

C2(5) (α) =

15α 4 (4

and 2 0 K (k)K (k), (10.6) π2 h n i p p 1 2 2 K 0 (k)K (k) + 4 (2 − α 2 ) + 2 E 0 (k)K (k) (α) = 4 − α 4 − α − α A(5) 1 π 2 α 2 (4 − α 2 )3/2 h o i p p − 4 (2 − α 2 ) − 4 − α 2 E(k)K 0 (k) − 8 4 − α 2 E 0 (k)E(k) , (10.7)

A(5) 0 (α) =

A(5) 2 (α) =

n ¡ ¢p 1 2 2 4 + 9α 4 − α 2 K 0 (k)K (k) − α 16 + 20α 12π 2 α 4 (4 − α 2 )7/2 i h p + 8 (64 − 56α 2 + 19α 4 − 6α 6 ) + (32 − 22α 2 + 11α 4 ) 4 − α 2 E 0 (k)K (k) i h p − 8 (64 − 56α 2 + 19α 4 − 6α 6 ) − (32 − 22α 2 + 11α 4 ) 4 − α 2 E(k)K 0 (k) o p (10.8) − 16(32 − 22α 2 + 11α 4 ) 4 − α 2 E 0 (k)E(k) ,

ON THE SIMPLE CUBIC GREEN FUNCTION

127

where k ≡ k(α) =

´1/2 p 1³ 2 − 4 − α2 2

(10.9)

(5) and α ∈ (0, 2). Recurrence relations for the coefficients A(5) n (α) and C n (α) can be obtained by substituting (10.1) in the differential equation (2.36). These relations and the initial conditions (10.6)–(10.8) and C0(5) (α) ≡ 1 enable one to generate exact formulae for the higher-order coefficients (5) {A(5) n (α) : n = 3, 4, . . .} and {C n (α) : n = 1, 2, . . .}. It follows from (10.9) and the discussion in Subsection 7(c) that if

α = α[N ] ≡ 4k[N ] (1 − k 2 [N ])1/2 ,

(10.10)

where k[N ] is the singular value of order N and N ≥ 2, then it is possible to express all the coefficients {A(5) n (α) : n = 0, 1, 2, . . .} in terms of just one complete elliptic integral K (k[N ]). A detailed analysis of the important case N = 3 with α[3] = 1 has been given by Joyce [16, p. 593]. Finally, we note that the integral representations (1.13) and (1.12) can be used to derive the (5) following alternative expressions for A(5) 0 (α) and C 0 (α), respectively, Z A(5) 0 (α) = G I (α, 0) = C0(5) (α)

∞

0

1 p = π α 4 − α2 2

J02 (t)J0 (αt) dt,

Z

∞ 0

t J02 (t)J0 (αt) dt,

(10.11) (10.12)

where α ∈ (0, 2). A comparison of these results with (10.6) and (10.3) yields the integration formulae Z

∞

0 ∞

Z 0

J02 (t)J0 (αt) dt =

2 0 K (k)K (k), π2

(10.13)

t J02 (t)J0 (αt) dt =

2 , √ π α 4 − α2

(10.14)

where k = k(α) is defined in (10.9) and α ∈ (0, 2). The formula (10.14) is in agreement with the work of Watson [27, p. 411], while the result (10.13) appears to be new. (b) Expansion for G(α, w) about w = 0 for α ∈ (2, ∞) When 2 < α < ∞ one finds that the expansion for G(α, w) about w = 0 can be written in the simple from G(α, w) = i

∞ X

2n A(6) n (α)w ,

(10.15)

n=0

where −π < arg(w) < 0 and |w| ≤ α − 2. The coefficients {A(6) n (α) : n = 0, 1, 2, . . .} are real functions of the parameter α, provided that α ∈ (2, ∞). It follows, therefore, that G R (α, u) ≡ 0, G I (α, u) =

∞ X n=0

where 2 − α ≤ u ≤ α − 2 and α ∈ (2, ∞).

(10.16) 2n A(6) n (α)u ,

(10.17)

128

DELVES AND JOYCE

Formulae for the coefficients {A(6) n (α) : n = 0, 1, 2} in the expansion (10.15) are 4 (10.18) K 2 (k), π 2α ³ ´ i h p p 2 2 2 − 4 E(k)K (k) + 2 α 2 − 4E 2 (k) , (10.19) A(6) (α) = (k) + 2 α − α −α K 1 π 2 α(α 2 − 4)3/2 i n h p 1 4 2 2 2 2 − 4 K 2 (k) + 28α − 8) − (α − 4)(7α − 8) α − 2α(α A(6) 2 (α) = 6π 2 α 3 (α 2 − 4)7/2 h i p + 4 α(α 4 + 28α 2 − 8) − (11α 4 − 22α 2 + 32) α 2 − 4 E(k)K (k) o p (10.20) + 4(11α 4 − 22α 2 + 32) α 2 − 4 E 2 (k) , A(6) 0 (α) =

where ´1/2 p 1 ³ α − α2 − 4 k ≡ k(α) = √ 2α

(10.21)

and α ∈ (2, ∞). A recurrence relation for the coefficient A(6) n (α) can be derived by substituting the series (10.15) in the differential equation (2.36). This relation and the initial conditions (10.18)–(10.20) enable one to generate exact formulae for the higher-order coefficients {A(6) n (α) : n = 3, 4, . . .}. We find from (10.21) and the discussion in Subsection 7(c) that if α = α[N ] ≡

1 , k[N ] (1 − k 2 [N ])1/2

(10.22)

where k[N ] is the singular value of order N and N ≥ 2, then it is possible to express all the coefficients {A(6) n (α) : n = 0, 1, 2, . . .} in terms of just one complete elliptic integral K (k[N ]). It should also be noted that the integral representation (1.13) can be used to express the coefficient A(6) 0 (α) in the form Z A(6) 0 (α)

= G I (α, 0) =

∞

0

J02 (t)J0 (αt) dt,

(10.23)

where α ∈ (2, ∞). If this result is compared with (10.18) we obtain the integration formula Z

∞ 0

J02 (t)J0 (αt) dt =

4 K 2 (k), π 2α

(10.24)

where k = k(α) is defined in (10.21) and α ∈ (2, ∞). The formula (10.24) is in agreement with the earlier work of Bailey [43]. (c) Expansion for G(α, w) about w = 0 for α = 2 The remaining task is to analyse the behaviour of G(α, w) in the neighbourhood of w = 0 when α = 2. For this special case we find that the expansion for G(α, w) can be written in the form ·X ¸ ∞ ∞ ∞ w 1/2 X w1/2 X n (1) n (7) 2n (1) n (−1) Cn (−2)w + i An (2)w − C (−2)w , G(2, w) = 2π n=0 2π n=0 n n=0

(10.25)

ON THE SIMPLE CUBIC GREEN FUNCTION

129

where −π < arg(w) < 0, |w| ≤ 2 and the coefficient Cn(1) (−2) is defined by Eq. (7.38). Formulae for the coefficients {A(7) n (2) : n = 0, 1, 2, . . .} can be obtained using the three-term recurrence relation 2 (7) 64n(n + 1)(4n + 1)(4n + 3)A(7) n+1 (2) − 8n(2n + 1)(20n + 1)An (2)

+ (2n + 1)(2n − 1)3 A(7) n−1 (2) = 0

(n = 1, 2, . . .)

(10.26)

with the initial conditions 2 2 K (k[1]), π2 ¸ · 1 3π 2 (7) 2 , 4K (k[1]) − 2 A1 (2) = 96π 2 K (k[1]) A(7) 0 (2) =

√ where k[1] = 1/ 2 is the singular value of order one. Finally, we note that the result (10.27) yields the further integration formula µ ¶ Z ∞ 2 1 J02 (t)J0 (2t) dt = 2 K 2 √ . π 2 0

(10.27) (10.28)

(10.29)

11. APPLICATIONS In this final section we shall briefly discuss some applications of the Green function G(α, w) in lattice statistics. (a) Lattice Dynamics The lattice dynamics of a set of N 3 identical particles which interact with nearest-neighbour interactions on an N × N × N simple cubic lattice has been investigated in considerable detail by Montroll [22]. In particular, he showed that for large N the frequency spectrum g(ν) of the model can be written in the form ¡ ¢ g(ν) = 4π ωN 3 Gˆ γ1 , γ2 , γ3 ; ω2 ,

(11.1)

where ¡ ¢ 1 Gˆ γ1 , γ2 , γ3 ; ω2 = π

Z

∞

cos[(ω2 − 2γ1 − 2γ2 − 2γ3 )x]J0 (2γ1 x)J0 (2γ2 x)J0 (2γ3 x) dx

(11.2)

0

and ω = 2πν. The parameter γ1 denotes the central force constant while γ2 , γ3 represent the non-central force constants. The maximum cut-off frequency for the model is given by ωL2 = 4(γ1 + γ2 + γ3 ).

(11.3)

If we restrict our attention to the special case γ1 = αγ , γ2 = γ3 = γ and compare (11.2) with the integral representation (1.13) we obtain the basic relation 2 ˆ ωL2 G(αγ , γ , γ ; ω2 ) = (2 + α)G I (α, u), π

(11.4)

130

DELVES AND JOYCE

where ¤ £ u = (2 + α) 1 − 2(ω/ωL )2

(11.5)

and ωL2 = 4γ (2 + α). The relation (11.4) and the results given in Sections 7–10 enable one to ˆ determine the exact behaviour of the spectral density function G(αγ , γ , γ ; ω2 ) at low frequencies and in the neighbourhood of all the Van Hove singularities. In particular, the low-frequency expansion ˆ for G(αγ , γ , γ ; ω2 ) may be readily derived by applying (7.5) to (11.4). It is found that ˆ ωL2 G(αγ , γ , γ ; ω2 )

2(2 + α)3/2 = √ π2 α

µ

ω ωL

¶X ∞

µ Cn(1) (α)[−2(2

+ α)]

n=0

n

ω ωL

¶2n ,

(11.6)

where the coefficient Cn(1) (α) is defined Eq. (7.38). It also possible to calculate the numerical value ˆ , γ , γ ; ω2 ) to great accuracy for any value of ω/ωL ∈ (0, 1) by using the elliptic integral of ωL2 G(αγ formula (5.13) and the procedure described in Subsection 5(c). (b) Spin-Wave Theory Consider the general spin S Heisenberg model of ferromagnetism for the simple cubic lattice with a nearest-neighbour exchange integral which has the values J, J , and α J in the x, y, and z directions, respectively, where J > 0 and α > 0. According to ideal spin-wave theory [44, 45] the magnetization M(T ) of this model at a thermodynamic temperature T is given by 1 M(T ) = 1 − Ä(α, τ ), M(0) S

(11.7)

where 1 Ä(α, τ ) = 3 π

Z πZ πZ 0

0

π 0

dθ1 dθ2 dθ3 , exp[τ λ(α, θ)] − 1

λ(α, θ) = (2 + α) − cos θ1 − cos θ2 − α cos θ3 , 4J S τ = kB T

(11.8) (11.9) (11.10)

and kB is the Boltzmann constant. We can use the density-of-states function (1.15) to express the integral (11.8) in the alternative form Ä(α, τ ) =

1 π

Z

2+α

−2−α

G I (α, u) du, exp[τ (2 + α − u)] − 1

(11.11)

where G I (α, u) is defined in Eq. (1.7). The low-temperature behaviour of the relative magnetization may be investigated by first expanding the integrand in (11.8) as a geometric series in powers of exp[−τ λ(α, θ)]. In this manner, we find that Z πZ πZ π ∞ X 1 exp[−mτ λ(α, θ)] dθ1 dθ2 dθ3 . Ä(α, τ ) = π3 0 0 0 m=1

(11.12)

ON THE SIMPLE CUBIC GREEN FUNCTION

131

Next the standard result [27] 1 π

Z

π

exp(x cos θ ) dθ = I0 (x)

(11.13)

0

is applied to the formula (11.12), where I0 (z) is a modified Bessel function of the first kind. This procedure yields Ä(α, τ ) =

∞ X

exp[−m(2 + α)τ ] I02 (mτ )I0 (mατ ).

(11.14)

m=1

When τ À 1 we can use the asymptotic formula ex I0 (x) ∼ √ 2π x

µ 2 F0

¶ 1 1 1 , ; , 2 2 2x

(11.15)

as x → ∞, to write (11.14) in the form ¶¸2 µ ¶ · µ ∞ X 1 1 1 1 1 1 1 1 Ä(α, τ ) ∼ , ; , ; , √ 2 F0 2 F0 2 2 2mτ 2 2 2mατ (2πτ )3/2 α m=1 m 3/2

(11.16)

as τ → ∞. Finally, the application of the formal identity (7.36) to (11.16) gives the low-temperature expansion Ä(α, τ ) ∼

¶µ ¶ µ ∞ µ ¶ X 1 n 3 3 1 (1) − C (α) ζ n + , √ 2 τ (2πτ )3/2 α n=0 2 n n

(11.17)

as τ → ∞, where ζ (s) denotes the Riemann zeta function and the coefficient Cn(1) (α) is defined by the Green function expansion (7.1) and the formula (7.38). (c) Random Walk Theory We conclude this section with an analysis of the random walk generating function R(t1 , t2 ) =

∞ X

2j

r (2i, 2 j) t12i t2 ,

(11.18)

i, j=0

where r (2i, 2 j) is the number of random walks on the simple cubic lattice which have a total of 2 j nearest-neighbour steps in the x or y direction and 2i nearest-neighbour steps in the z direction, and which also return to the starting point (not necessarily for the first time) at the end of the walk. (It has been assumed that the cubic axes of the lattice are in the x, y, and z directions.) From the work of Montroll and Weiss [7] it can be shown that 1 R(t1 , t2 ) = 3 π

Z πZ πZ 0

0

π 0

dθ1 dθ2 dθ3 . 1 − 2t2 (cos θ1 + cos θ2 ) − 2t1 cos θ3

(11.19)

If this result is compared with (1.1) we see that R(t1 , t2 ) =

1 G 2t2

µ

¶ t1 1 , . t2 2t2

(11.20)

132

DELVES AND JOYCE

It follows, therefore, that we can use the basic formula (5.13) to express the generating function R(t1 , t2 ) in terms of complete elliptic integrals of the first kind. Next we apply the moment expansion (2.1) to Eq. (11.20). This procedure yields

R(t1 , t2 ) =

∞ X n=0

¡1¢ 2 n

(1)n



−n,

 (2t1 )2n 3 F2  

1,

−n,

1 ; 2

  (2t2 /t1 )2  .

(11.21)

1;

It is now possible to derive a formula for r (2i, 2 j) by rearranging the expansion (11.21). In particular, we find that ¶ µ ¶2 µ 22i+4 j 1 1 r (2i, 2 j) = ( j + 1)i . j+ (i!)2 ( j!)2 2 j 2 i

(11.22)

Finally, we follow the procedure described by Guttmann and Enting [46] and write the generating function (11.18) in the alternative form R(t1 , t2 ) =

∞ X

2j

H j (t1 ) t2 ,

(11.23)

r (2i, 2 j)t12i .

(11.24)

j=0

where H j (t1 ) =

∞ X i=0

The substitution of (11.22) in (11.24) gives 24 j H j (t1 ) = ( j!)2

µ ¶2 µ ¶ 1 1 2 2 F1 j + , j + 1; 1; 4t1 . 2 j 2

(11.25)

If the standard formula [47, p. 562] ¶ ¶ µ µ 1 1 −a , P2a−1 √ 2 F1 a, a + ; 1; z = (1 − z) 2 1−z

(11.26)

where Pn (x) denotes a Legendre polynomial, is applied to (11.25) we see that   q 4 j µ ¶2 2 1 1 1 . H j (t1 ) 1 − 4t12 = ¡ ¢ P2 j  q ( j!)2 2 j 1 − 4t 2 j 1 − 4t12 1

(11.27)

√ It is readily found from (11.27) that H j (t1 ) 1−4t12 is a rational function of t1 with poles of order 2 j at t1 = ±1/2. This result provides one with a simple example of the solvability criterion introduced by Guttmann and Enting [46].

ON THE SIMPLE CUBIC GREEN FUNCTION

133

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