Chapter 13 Multiple Sequences

Chapter 13

Multiple Sequences

13.1. Rectangular Transformations

Given a double sequence {snd, define the transformed double array

{snd componentwise by

n,k 2 0.

(1)

The transformation is completely characterized by the four-dimensional array of weights U = [p7f], s i :s;; n, o « j :s;; k. (2) Obviously, convergence in the {snd array can be path dependent. Let Snk be located at the point (k, -n) of JO x - J . We shall be concerned here with only two modes of convergence, horizontal, limk _ oo Snk, and vertical limn _ oo Snk' (This designation differs slightly (rom the convention previously used for array transformations s -> S(k l, but here it is more useful to think of {Snk} as a rectangular, rather than a triangular, array.) We shall assume that

°

lim Snk =

k-

fln,

n 2 0,

00

k 20. Definition. regular if

(3)

The transformation defined by (1) is called horizontally lim

k-oo

Snk

=

fln'

227

n 20,

(4)

228

13. Multiple Sequences

and vertically regular if k

~

(5)

0.

The material in the remainder of this section is due to Higgins (1976). Theorem 1. The transformation defined by (1) is horizontally regular iff (i) given n

0,

~

n

k

L L IlliY I ~ R n

;=0 j=O

for some positive number R; independent of k; (ii) given n ~ 0, lim

°

k

L 1l~1 =

bnr ;

k-oo j=O

(iii) given n

0, j

~

~

0,

n,

~ i ~

lim lliY = 0.

k-oo

Proof <=: By virtue of (3), one can write sij = fii

Thus

Snk - fin

=

n

+ £ij' where

lim j _

oo

£ij = 0.

k

L L Ili'f(fi; + £i) -

;=0 j=O

fin

(6)

For n fixed, condition (ii) guarantees that the first two terms on the righthand side of (6) can be made arbitrarily small for large k. Now separate the remaining term as n

k

n

n

J

L L Ilijk£ij = L L lliY£ij + L

;=0 j=O

i=O j=O

° ° °

k

L lliYeij'

i=O j=J+ I

(7)

°

Since n is fixed and lim j_ 00 eij = for ~ i ~ n, given s > it is possible to pick J such that IeijI < e/2R n for ~ i ~ n andj > J. Thus for k sufficiently large, the second term on the right-hand side of (7) has modulus less than e/2 when condition (i) applies. Now with nand J fixed, define M

=

max leijl,

O::s;isn

o ,.:,j,.:,J

(8)

13.1. Rectangular Transformations

229

and pick K so large that k > K implies

°

Ifl71 I < c/2M nJ

°

(9)

for ~ i ~ nand ~ j ~ J. Condition (iii) has obviously been used to guarantee (9). Thus for k sufficiently large, both terms on the right-hand side of (7) can be made arbitrarily small so that the three conditions of the theorem imply limk~GO (Snk - f3n) = for each n. =>: First, let r be fixed and apply the transformation to the array

°

I,

Snk

obtaining

a, =

-

Snk

{

°

n = r, k 2 otherwise

= { 0,

(10)

r> n

k

fl,)'

"nk

L...

j~O

(11)

n 2 r.

By the horizontal regularity,

l5 n, = lim Snk = lim

k

I

(12)

n 2 r.

fl~J,

k-r sx» j==O

k-oo

Now let r vary to demonstrate the necessity of condition (ii). Second, for the necessity of condition (iii), fix i andj and define the double arrays Dij = (dnk ) , where dnk = l5 inl5 j k • Applying the transformation, we obtain n 2 i, k 2j otherwise.

Thus for i

~

n, · 1im

k-co

flijnk

- = liim = liim Snk k-rJ:)

k-oo

Snk

=

(13)

°

(14)

by horizontal regularity. Varying i andj, we obtain the necessity of condition (iii). Third, the necessity of condition (i) will be demonstrated by contradiction. Suppose there is an integer n for which k

n

lim k~GO

I I

i~O j~O

Ifl71 I = + 00.

Then there must be at least one integer i such that k

lim

I

1

k~GOj~O

fl71 1 = + 00.

°

~ i ~

(15)

nand (16)

230

13. Multiple Sequences

If i = n, define the Toeplitz transformation with matrix (c k ) by k 2 0,

O:::;;j:::;; k.

(17)

This Toeplitz matrix represents a nonregular method since L~ 1 Ickj I is unbounded. Thus, there is a sequence {x) with lim, Xj = x and L'=o CkjXj does not converge to x as k goes to infinity. Consider now the double array that has zero entries except for row n, wherein lies the sequence {xj}, and apply the transformation of the theorem to obtain as

k

~

co.

(18)

Therefore the method is not horizontally regular if i = n. If i < n, consider the Toeplitz transformation whose matrix (Ckj) is given by nk k 20, O:::;;j:::;; k. (19) Ckj = J1ij' By Hardy (1956, p. 43), k

L Ickjl = + co k-oo j=O fiiii

implies the existence of a bounded sequence {y i} with the property that {L'=o CkjYj}k"= 1 does not converge as k ~ co. Now apply the transformation of the theorem to the double array (Snk), which has all zero entries except in row i, wherein lies the sequence {yJ. Clearly, (20) but limk_oo Snk = 0; thus if i < n, the method is not horizontally regular. The necessity of condition (i) is now established by contradiction. • The proof actually substantiates some stronger results regarding the transformation (1). For example, note that with n fixed, the three conditions of the theorem are necessary and sufficient for lim k_ oo Snk = Pn. Also, conditions (i) and (iii) are sufficient conditions that the transformation (1) map a double array with all row limits zero to a double array with all row limits zero. The three corresponding necessary and sufficient conditions for vertical regularity can be obtained by analogy. The existence of higher-dimensional analogs of these methods and of this theorem also is clear. An extension of these ideas that has practical application but will not be pursued in this book is the following.

231

13.1. Rectangular Transformations

Suppose f: JO -+ JO and g: JO -+ JO with either limn fen) = limn g(n) = 00 (or both). Call the double array (f, g)-convergent to s if lim

=

S!(n). g(n)

00

or (21)

S.

The natural question is, What are necessary and sufficient conditions on the weights Ili} in (1) to ensure that if (Snk) is (j, g)-convergent to S then (snd is (j, g)-convergent to s? If the double array (Snk) has lim Snk k~oo

= /3,

n 2:: 0,

(22)

then condition (ii) of Theorem 1 can be somewhat relaxed and still maintain horizontal regularity of the transformation (1).

Theorem 2. Suppose that the double array (Snk) enjoys property (22) and that conditions (i) and (iii) of Theorem 1 are satisfied. If n

lim

k-v co

then

n

lim

k-r o:

Proof

L ;;0

k

'Illi}

=

1,

n 2:: 0,

(23)

lliJSij

= /3,

n > 0.

(24)

j;O

k

I I ;;0

j;O

Obvious in view of the proof of Theorem 1.

•

This theorem aids in the design of transformations of the double arrays. If we assume that the better approximations to the limit /3 appear for the larger indices nand k, the weights Ili} should put more mass on the larger indices i and j than on the smaller indices. Therefore, let fbe a function from JO x JO to P/I+ satisfying f(Il,j) > f(v,j)

for

11 > v

and all j,

(25)

fU,Il) > f(i, v)

for

11 > v

and all

(26)

i,

and 'I);O f(O,j) diverges. We choose the weights Ili} by Ili}

= f(i,j) l,to

.t

o

f(ll, v),

(27)

which leads to 'Ii;o 'I); 0 Ili} = 1 for all nand k, so that (23) is satisfied. Condition (iii) of Theorem 1 is satisfied in view of the divergence of the sum 'I); 0 f(1,j)· Condition (i) is satisfied because of the positivity off Thus, the weights (27) define a transformation for which limk~ 00 Snk = P for all n implies lirn.., 00 snk = /3 for all n. These weights will be computationally useful only when the functionfreflects the input double array.

232

13. Multiple Sequences

13.2. Crystal Lattice Sums

An important class of multidimensional sums arises in the theory of crystal lattices, specifically in the computation of the lattice energy per atom of a given crystalline material. Let f: JP -+~, JP = J x ... x J, M P = (ml"'" mp), Ap=(al"'" ap), Op=(O, 0, ... ,0), IIMpl1 =(mi + ... + m;)1/2. [Where there is no chance of misunderstanding, we omit the subscript p.) The sums of interest are generally of the form S=

M~A

f(M)

11M - A11 2 s '

S E

CfJ.

(1)

f is usually quite simple, typical examples being f

= 1,

(2)

although in so-called phase modulated sums (Glasser, 1974) more complicated functions occur. Sometimes the mj range over only even or odd numbers, but it is not useful to develop a special notation to deal with such cases. It is not at all obvious when (1) converges. The following theorem is often applicable. Theorem. Let M - A#-O and let f be bounded. Then (1) converges and represents an analytic function of s for Re s > p/2, convergence being obtained regardless of the order in p-space in which the terms are added up.

Proof The most elegant demonstration uses the theory of theta functions. This proof is given in Section 13.2.2. •

For a discussion of the physical context in which such sums arise, see the classic treatise by Born and Huang (1954). We shall take an approach with these sums that is fundamentally different from the procedures used previously in this book to accelerate the convergence of series or sequences. The techniques given here will not be general, but will very much depend on the specific character off This is, of course, very much in contrast to the previous work-for instance, the fact that the remainder sequence possessed an asymptotic series of Poincare type-where only the general form of the sequence or series was of interest. The present kind of endeavor might be called the analytic approach to sequence transformations. The arguments used will depend on known properties of mathematical functions, such as theta functions, and on the application of a powerful formula from classical analysis, the Poisson summation formula.

13.2. Crystal Lattice Sums

233

13.2.1. Exact Methods Definition.

Let f be locally L(O, 00) and let the integral .A(f; s) = {"" x·- 1f(x) dx

(1)

converge for Re s = to, Re s = t 1, to < t t- .A is called the Mellin transform

off

Clearly the integral converges for IX

< Re s < {3

(2)

where IX = inf to and {3 = sup t l' (2) is called the strip of absolute convergence of (1). The Mellin inversion theorem states that iffis of bounded variation in a neighborhood of x E (0, 00), then, for any IX < C < {3, f(x+)

+

f(x-) _ _ 1 l' fC+iR «cj, ) . 1m JI't, S X 2m R~"" c-iR

'---'-----'--=--'-----'- -

2

-r

s

d

S,

(3)

Usually.A may be continued analytically into a larger region q; of the complex s plane, for instance, f(s)

-. = a

f"" x

.-1

0

e

-ax

d

X,

Re a> 0.

(4)

°

Here IX = 0, {3 = 00, q; = Cfi - {a, -1, -2, ... }. The theta functions for x > are defined as follows.

(5)

For some of the many beautiful properties of these now almost forgotten functions, consult Whittaker and Watson (1962), Hancock (1909, Vol. I), or Bellman's more recent book (1961), which is compulsively readable. A good collection of formulas is in Abramowitz and Stegun (1964). The following notation is standard:

B/O, q) = BlOlr),

(6)

234

13. Multiple Sequences

Thus 8i(0Iix/n) corresponds to taking q = e- x in 8i(0, q). Formulas such as the following can be found in Hancock (1909, Chapter XVIII):

(7)

Similar formulas exist for OJ, etc.; see Hancock or Jacobi (1829), who gives a list of 47 such relationships. It can be shown that 8/0Iix/n) has an algebraic singularity at x = 0; hence Mellin transforms of Oz, 8 3 - 1, 84 - 1, etc., have a half-plane of convergence. The Mellin transforms of theta functions generally involve meromorphic functions such as Riemann's zeta function, defined for Re s > 1 by (s)

=

1

L s' n 00

(8)

n: 1

We shall need the formulas (1 - 2 1 - ' )( s)

=

00

(_1)"-1

n: 1

n

L

"

Re s > 1; (9)

Re s > 1. Another useful function is Re s > 1,

(10)

which satisfies the relationship L(1 - s)

= (2/n)'r(s) sin(ns/2)L(s).

(11)

Obviously, L(s) can be expressed in terms of the generalized zeta function 00

(s, a)

1

= n:O L (n + a)S'

-a ¢ JO,

Re s > 1.

(12)

13.2. Crystal Lattice Sums

235

The Mellin transforms of powers of the theta functions can be found from such formulas as (7). For instance,

A[8~(01~)J = 4AL~o(-lte-(n+1/2)(2k+l)XJ (-It

00

= 4r(s)

n.~o (n + t)S(2k + I)' 2s

00

= 4r(s)L(s)n~o (2n + I)S = 4(2

S -

1)r(sK(s)L(s). (13)

Mellin transforms of products of theta functions can be found by using the

Landen transformations,

8iO, q)83(0, q)

= ¥1~(0, ql/2),

8iO, q)8iO, q) = te-"i/48~(0, i q I / 2),

(14)

83(0, q)8iO, q) = 8i(0, q2),

and the formula

A{f(ax); s} = a-sA{f(x); s}.

(15)

Table I gives some of the Mellin transforms that can be found this way. Table I Mellin Transforms Involving OJ

=

Ii/Ol ixln)

f(x)

O2

2(2 2 '

04

2(2 1 - 2 ' - 1)[(s)(2s) 4(2' - 1)r(s)(s)L(s)

03 O~ O~

O~

(0 3 (0 4

- I

l)r(s)(2s)

1)2 1)2

_

_

0 304 -

-

I 1)(04

0 20 304 0'2

OJ - I

01-

4r(s)(s)L(s) 1)[(s)(s)L(s) 4r(s)[L(s)(s) - «2s)] 4(1 - 2 1 - 2')[(S)[(2s) - L(s)(s)] 2>+ [(2' - 1)[(s)(s)L(s) 22-'(2 1 - ' - l)f(s)(s)L(s) -2 2 - ' [(s )[r ' ( 2s ) + (I - 2 1-')(s)L(s)] -2'+ [r(s)L(2s - I ) 16(1 - 2'-')(1 - 2-')[(s)(s)(s - I)

4(2 1 - ,

- I

020 3

(0 3

-

2[(s)(2s)

I

O~O~ O~O~ - I 8~0~

-

I)

8(1 - 2 2 - 2')f(s)(s)(s - I) - 8(1 - 21-»(1 - 2 2 -')f(S)(5)(S - I) 2>+ 2f(s)L(s)L(s - I) _2 3-'(1 - 2 2 - ' )(1 - 2[-')r(s)(s)(s - I) 2 2 +' (1 - 2 1 - 5 )(1 - 2-')f(s)(s)(s - I)

236

13. Multiple Sequences

To see how these formulas can be used to obtain closed-form expressions for lattice sums, consider

~, = L,

1 r(s)

--

-00

= - 1

r(s)

foo x

e

s-1 -(m 2+m 2+m 2+m 2)x

0

I

2

foo xS-1[e~(Olix/n) 0

3

4

dX

1] dx

= 8(1 - 22 - 2 S)((S)((s - 1).

(16)

[Later it is shown that this sum converges for Re s > p/2 = 2. Since ((s - 1) has a pole at s = 2, the result is sharp.] As another example, consider

= L(s)((s)

(17)

- ((2s).

A short table (Table II) lists two-dimensional sums determined by Glasser. Table II cc

S =

L' J(m,n)

J

S

(m 1 + n 2 ) - ' (_l)m+n(m 1 + n 2 ) - ' (_I)n+ '(m 2 + n 2 ) - ' [(2m + 1)2 + (2n + 1)2r'. m, n Z 0

4«s)L(s) -4(1 - 2'-2')(s)L(s) 22-'(1 - 2'-')(s)L(s) 2-'(1 - r')(s)L(s) 2(1 - 2-' + 2'-1')(s)L(s)

(m 2

+ 4n 1 ) - '

13.2. Crystal Lattice Sums

237

Certain other related sums have been obtained, i.e.,

L' (m + mn + n ) - S = 6(s)g(s), 00

2

2

g(s)

=

-00

I

00

n=O

[(3n

+ 1)-S - (3n + 2)-S] (18)

(Fletcher et al., 1962, p. 95), and (19) whose derivation is rather complicated (Glasser, 1973b). Obviously, the following case can be expressed by a single sum:

L (ml

mj?l

+

m2

+ ... +

mp )

-r

s

=

L (k + Pk 00

k=O

1) (k 1 y' +P

(20)

The difficulty in computing odd-dimensional sums by the use of theta functions is that most of the known theta function identities involve an even number of theta functions. Glasser (1937b) uses a number-theoretic approach to obtain additional sums, and the theory of basic hypergeometric series (Glasser, 1975) can be used to deduce the five-dimensional sum

L

ml?:O;m2,"';m5~

(m 1m2 + m1m3 + m3m4 + m4ml + m2mS)-S

1

= (S)(S - 2) - (2(S - 1). (21)

(The region of convergence of this sum cannot be deduced from the theorem of Section 13.2.) 13.2.2. Approximate Methods: The Poisson Summation Formula

Many approximation techniques have been developed to deal with lattice sums, beginning, perhaps, with Born's and Huang's approach, which uses values of the incomplete gamma function. That approach is not very adaptable to general values of s. Other approaches (van der Hoff and Benson, 1953; Benson and Schreiber, 1955; Hautot, 1974) use methods that convert the sum to a multidimensional sum involving the modified Bessel functions K v • This might, at first glance, seem to be compounding the problems. However, the transformed sums converge with extraordinary rapidity, and often the contributions at just a few lattice points serve to give six- or eightplace accuracy. Several approaches are possible, including one (Hautot, 1974)using Schlornilch series. My own preference is to begin with the following striking result, which can be found in any book on Fourier methods [e.g., Butzer and Nessel (1971, p. 202)].

238

13. Multiple Sequences

Let f

Theorem.

E

L( -

F(x) =

00,

(0),

Loooo e-iX~f(t)

dt,

X E

(1)

!Jll.

Then, iff is of bounded variation, 2n

L 00

k=-oo

f(x

+ 2kn) =

lim

n

X E

n-cok=-n

where, at points of discontinuity, f(a)

Proof

L eikxF(k),

= -t[f(a+) +

See Butzer and Nessel (1971).

211,

(2)

f(a-)].

•

There follows a list of formulas that will subsequently be of use. For the computation of the integrals involved, consult Erdelyi et al. (1954, Vol. I). f(t) = e- at2 cos bt, a E g~+, b e .OJ;

f

e- a(x+2kn)2 cos[b(x

+ 2kn)J

= _1_

- 00

f

2J"1W -

00

eikxe-(k2+b2)/4a COSh(bk). (S-l) 2a

f(t) = Itl±IlK(altJ),

f

eikX(k 2 + a 2)+11-1/2 =

-00

a E!Jll+;

2J1r + 1/2)

(S-2)

(2a)±Ilr(±/l 00

x

L

[x

-00

+ 2knl±IlKialx + 2knl).

(By analytic continuation and use ofthe well-known asymptotic properties of K Il , one finds that these sums are convergent and equal when Re( ± /l) > 0.) f(t) = Jt 2 + a 2 - l e - b-/tT +a' , a, b E ,OJ+; n

L 00

L 00

[(x + 2kn)2 + a2r1/2e-bv'{.x+2kn)2+a2 =

-00

-00

eikxK o(aJb 2 + k 2); (S-3)

a, b E .0/1.+ ;

L 00

~ e- b-/(;:;: 2kn)'+-a2 ab_ oo

L eikxJb 2 + k 2- 1 00

=

-00

x K 1(aJb 2 + k 2);

(S-4)

a, b E ]1+;

}br.3 a ±I'b 1/H Il L 00

-00

L 00

=

-00

[(x

+

2kn)2

+ a 2J±Il/2-1/4K±I1_ 1/z(bJ(x+2kn)2+ a2)

eikx(k2 + b 2)+11/2K,,(aJb 2

+ e),

/l E

t.{j.

(S-5)

239

13.2. Crystal Lattice Sums

We are now in a position to complete the proof of the theorem in Section 13.2. Let (3) (Without loss of generality we may assume that A = 0.) Then

S

=

9=

1

r(s)

Jo 9 dx, (00

Xs- 1

L

(4)

f(M)e-IIMII2x.

(5)

Imjl
We get

IJ =

9 ~ CXReS-l[83(0Iix/n)p and, by (S-I) with b

h,

(6)

x =0

=

h = O(xRes-l-PIZ),

X

--+

O+,

(7)

while as x --+ 00, h = O(e- ax ) , a > O. Thus, under the stated conditions, h is integrable and, by dominated convergence, limNr'oo Sexists. • Expansion (S-2) will be the principal tool. Let k a =

multiply both sides by and sum:

f

00

[IIMpll z

+ JZJs

23/Z-S~

=

r(s)

-

Il,

f

00

take the upper sign on [x

+ 2m

--+ O.

fl, fl--+ S -

t,

nls-l/ZeiX(m,+· .. +mp-Il

[IIM;_ll1 z

X KS-l/z
mp ,

(mi + ... + m;-l + ()Z)l/Z,

eix(m'+"'+m p

eix(m,+ .. ·+m p)

-+

+ JZJS/Z 1/4 + JZlx + 2mpn l).

.

(8)

The result can be written [x

+ 2m nl s- 1/ z 11M p_lll s - 1/ Z

"---_ _~p_~_ eix(m,

x KS-l/z(IIMp-ll1lx

00

+ ... + mp -

i )

cos kx

+ 2mpn l) + 2k~I~'

As it stands, this holds only for x '# 2jn, j term must be peeled off and the relationship

E

J. For x

--+ 0+,

the mp

(9)

=0 (10)

240

13. Multiple Sequences

used. The result gives the original sum as a sum over one lower dimension plus a rapidly convergent series of Bessel functions.

~ IIMpll-2s = 2(2s) + y0tr~(S~ 1/2) +

2n s r(s)

f:

IIM p_t11 1- 2s

1m Is-I/2

!;: IIMp:llIs 00

1/2 Ks_I/2(IIMp_11112mpnl).

(11)

The Besselfunction expansions on the right of (9) and (11), still expansions over p-space, converge with great rapidity. Also, for the values of x of greatest interest, the cosine series on the right of (9) can be evaluated in terms of zeta and related functions. For other values of s, it can be dealt with by the asymptotic techniques of Section 1.6. In many cases, s is an integer. The series on the right then becomes a series of exponentials. (An example is given later on.) In any event, the Bessel function K; can be considered a known quantity, its computation today being standard software. For s = t in the case of a three-dimensional sum, there is convergence provided x '1= 2jn, j E J. The (m l , m2) sum can be expressed in terms of exponentials by (S-3), i.e.,

L 00

L 00

mt=l

eiX(m 1 +m,) Ko(aj

mi + mD = n L 00

m2=-OO

m2=-cIJ

x

[(x

+ 2m2n)2 + a2r

(eJ(X + 2m,n-j'-+ a L

ix _

1)-

I,

1/2

(12)

and this can be used in an obvious way in (11). The same applies, with (S-4), when p = 3 and s = l As an example of how an error analysis of these sums proceeds, let us examine (11). Assume the Bessel function sum is truncated, with all points inside the hypercube sup Ixjl = N

(13)

o s i s»

included. Let RN =

I

Imj[=N+I

1m Is - I/2 11M P II s- I/2 Ks_I/2(IIMp_11112mpnl). p-I

(14)

For an analysis of R N , we shall need several preliminary results useful with sums of this kind. Lemma 1. Let

IX,

n > 0, f3 > IX/n. Then

L kae- Pk :::; nae - Pn(1 00

k=n

ea/n-p)-I.

(15)

241

13.2. Crystal Lattice Sums

By calculus one finds that

Proof

x' Letting x

~

(rx/(jYe-ae bx ,

a, (j, x >

o.

k, (j --> «[n, and substituting the result in (15) proves the lemma.

-->

•

--!-,

For A ~ 1, Re v >

Lemma 2.

I

K.(A)eA r(Re v + 1) AV ~ jr(v + -!-)I KRe.(l) =

I Proof

(16)

A(

(17)

CV •

This follows immediately from the integral

Kv(z)e= -- = ZV

- r

v

2

+ -1)-1 2

Re v>

-1-,

foo e 0

-zt[t (I + -t)]V-I/2 dt

Re z >

2

'

o. •

(18)

Lemma 3 (19) Proof

(20) so (mi

and the lemma follows.

+ ... +

m;)1/2 ~ (l/p)(m l

+ ... + mp )

(21)

•

A straightforward application of all these results shows that for s ~

1< 2s+p+I/2rr2S-I/2cs_I/z{N + 1)2S-1 exp{-[2rr(p

R

I

N

r(s)

-

x {I - exp[ -2rr(N

+

2s + p+ 1/2 rr2s-I/2 r(s)

K s-

+

1)2}

1)/p]}I-P

x {I - exp[(2s - 1)/(N >::::

- 1)/p](N

-!-

+

1 / 2 ( 1)

1) - 2rr(p - l)(N

+

exp{ -[2rr(p - 1)/p](N

l)/p]}-I

+

1)2}, N

--> 00.

(22) For instance, if N = 2, the truncated sum will contain 26 terms if p = 3. The exponential term above is 4.2 x 10- 1 7 . If only seven terms are taken (N = I), the exponential term is still only 5.3 x 10- 8 .

242

13. Multiple Sequences

The case s = 1 of (9) is particularly important. It gives eix(m, + ... +mpl

v- 2 2=n -oomt+···+mp 00

'\' 00

L.

x

L.

eix(m'+"'+m p'

l )

-00

(ml ... ·.mp-,l"O

e-j;;'T+"'+~--;lx+2mp"l

00

cos kx

Jmi + ... + m;-t

k=t

k2

+ 2L

(23)

a rapidly convergent series of exponentials. Obviously the forgoing procedure is easily modified to account for sums with denominator 11M - Ails, A = (at, ... , a p ) . For many special cases, see Hautot's paper. 13.2.3. Laguerre Quadrature

This is an elementary but very accurate method for hand computations. It can be applied for certain functionsfwhen s - 1 - 1P is a value {3 for which the abscissas and weights for the Laguerre quadrature formula for xfJe- x have been tabulated, e.g., {3 = 0, -t, -1-. -1, etc (Concus et al., 1963.) This is illustrated for f == 1.

(1) h(x)

=

exx P/ 2[03(0Iix/ny - 1].

The integral on the right is easily evaluated by Laguerre quadrature, since the series for 0 3 converges with great rapidity. For example, let P = 2, s = l

(2)

Laguerre quadrature with just three abcissas yields S = 9.0352, while the true value is 9.0336.