A globally uniform asymptotic expansion of the hermite polynomials

Acta Mathematica Scientia 2008,28B(4):834–842 http://actams.wipm.ac.cn

A GLOBALLY UNIFORM ASYMPTOTIC EXPANSION OF THE HERMITE POLYNOMIALS∗ Shi Wei (


)

School of Mathematics and Statistics, Wuhan University, Wuhan 430071, China E-mail: [email protected]

Abstract In this article, the author extends the validity of a uniform asymptotic ex√ pansion of the Hermite polynomials Hn ( 2n + 1α) to include all positive values of α. His method makes use of the rational functions introduced by Olde Daalhuis and Temme (SIAM J. Math. Anal., (1994), 25: 304-321). A new estimate for the remainder is given. Key words Hermite polynomials, uniform asymptotic expansion, Airy function 2000 MR Subject Classification

1

41A60, 33E20, 33E20

Introduction

The Hermite polynomials Hn (x) is undoubtedly one of the most frequently used orthogonal polynomials in mathematics. Classical results on the asymptotic behavior of this polynomial can be found in Szeg˝ o’s definitive book Orthogonal Polynomials [8]; among them, an interesting √ and important result is the formulas of Plancherel–Rotach for Hn ( 2n + 1α) [8, p.201]. But, a far more satisfactory result was later given by Skovgaard [6], which combines all three formulas of Plancherel–Rotach into a single one, and leaves no gap in the intervals of validity. Because Hermite polynomials are related to the parabolic cylinder function, in this connection we must also mention the works of Erd´elyi [3] and Olver [5]. However, all their results were derived from the Hermite differential equation, and made use of the turning-point theory developed for second-order linear differential equations. As far as we are aware, there was only one article before 1990 on the asymptotic behavior of the Hermite polynomials, which was based on an integral representation. The article was by Wyman [9], and his result is considerably less global in nature than those obtained via differential equation theory. In a more recent article [7], Sun derived an asymptotic expansion by using a contour integral. His method is based on the cubic transformation discovered by Chester, Friedman, and Ursell [2], and on a repeated application of an integration-by-parts technique introduced by Bleistein [1]. The only problem with Sun’s result is that the validity of his expansion is limited to finite values of α, whereas according to Skovgaard [6] and Olver [5] this result should hold even when α goes to infinity. ∗ Received

April 14, 2006

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The objective of this article is to extend the validity of Sun’s expansion to include all positive values of α. Our approach is to apply a method of Olde Daalhuis and Temme [4] which was specifically designed for such purposes.

2

The Cubic Transformation The Hermite polynomial Hn (x) has the integral representation  2 n! −n N Hn (N α) = g(z)eN f (z,α) dz, 2πi Γ 1

where g(z) = z − 2 , f (z, α) = 2αz − z 2 − 12 log z, x = N α and N = are located at  1 z = z± = (α ± α2 − 1); 2

(1)

√ 2n + 1. The saddle points (2)

they coalesce at z = 12 when α = 1. For convenience, we consider two different cases: (1) 0 ≤ α < 1 and (2) α ≥ 1. The saddle points and their associated steepest paths are shown in Figure 1. In this figure, the arrows indicate directions of descent. The cubic transformation of Chester, Friedman, and Ursell [3], mentioned in Section 1, is defined by setting 1 1 2αz − z 2 − log z = w3 − b2 w + c. (3) 2 3 Upon differentiation on both sides, we get −

2(z − z− )(z − z+ ) dz = w 2 − b2 . z dw

(4)

For the transformation z → w to be one-to-one and analytic, we must make z = z+ and z = z− correspond to w = −b and w = b, respectively. This will lead to ⎧  3 ⎪ −1 ⎪ ⎨ i (cos α − α 1 − α2 ), if 0 ≤ α < 1; 4 b3 (α) = (5) ⎪ 3  2 ⎪ ⎩ (α α − 1 − cosh−1 α), if α ≥ 1, 4 and c(α) =

1 2 1 1 α + + log 2. 2 4 2

(6)

Note that from (5) we have b3 (α) ∼

3  2 α α −1→∞ 4

as α → ∞.

We now deform the contour Γ in (1) into the steepest descent paths shown in Fig.1. Changing the integration variable in (1) from z to w, we obtain  2 1 3 2 n! −n N 2 c Hn (N α) = N e eN ( 3 w −b w) h0 (w)dw, (7) 2πi L where 1

h0 (w) = z − 2

dz 1 1 (w − b)(w + b) = − z2 , dw 2 (z − z− )(z − z+ )

(8)

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and L is the image of the steepest paths shown in Fig.1. Note that a conformal map takes steepest descent paths into steepest descent paths. This fact is reassured by the graph of L shown in Fig.2.

(1)

(2) Fig.1

Steepest paths

(1)

(2) Fig.2

3

Contour L

An Integration-by-parts Technique

To derive an asymptotic expansion for the integral in (7) which holds uniformly for b near 0, Bleistein [1] introduced a clever technique of integration by parts. His method proceeds as follows. Define hm (w) = am + bm w + (w2 − b2 )ψm (w),

(9)

and hm+1 (w) =

d ψm (w), dw

(10)

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for m = 0, 1, 2, · · ·. The coefficients am and bm are easily determined by substituting w = ±b in (3.1). Inserting (9) in (7) and integrating by parts p times give 

p−1 p−1 2 1 Ai(N 4/3 b2 ) Ai (N 4/3 b2 ) m am m bm Hn (N α) = N −n eN c (−1) − (−1) + ε p , n! N 2m N 2m N 2/3 N 4/3 m=0 m=0 (11) where  2 1 3 2 (−1)p 1 εp = eN ( 3 w −b w) hp (w)dw, (12) 2p N 2πi L and hp (w) has the same domain of analyticity as h0 (w). The details of the above derivation are given in [7], where it is also shown that for α in a bounded interval [0, M ], there exist constants cp and dp , independent of N and b, such that |εp | ≤ cp

4

|Ai(N 4/3 b2 )| |Ai (N 4/3 b2 )| + d . p N 2p+2/3 N 2p+4/3

(13)

A Class of Rational Functions In [4], Olde Daalhuis and Temme introduced the sequence of rational functions R0 (u, w, b) = Rn+1 (u, w, b) =

1 , u−w

−1 d Rn (u, w, b), − b2 du

u2

(14) n = 0, 1, 2, · · · ,

(15)

where u, w, b ∈ C, u = w, and u2 = b2 . Furthermore, by using induction on p, they gave the representation Rp (u, w, b) =

kp,i p−1 i=0 j=0

(u −

Cij ui−j p+1−i−j w) (u2

− b2 )p+i

,

p = 1, 2, · · · ,

(16)

where kp,i = min(i, p − 1 − i) and Cij do not depend on u, w, and b. From (16), they deduced the following result. Lemma 1 Let w ∈ C be such that |w − b| = O(b) as b → ∞, and let γ be a simple closed contour that encircles b and w. Then for p = 1, 2, · · ·,  1 Rp (u, w, b)du = O(b−3p ) as b → ∞. (17) 2πi γ

5

An Estimate for hp(w)

To extend the validity of the asymptotic expansion given in (11) and (13) to unbounded values of α, we define as in [4] ρ0 (b) = min{|w ± b| : w is a singularity of h0 (w)}.

(18)

√ For α ≥ 1, the points z± = 12 (α− α2 − 1) in the z-plane are mapped into w = ∓b, respectively, under the mapping z → w given in (3), when the logarithmic function is restricted to its

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principal value. However, the points z± on other sheets of the Riemann surface of logarithmic function are singularities of the mapping (3). π Lemma 2 As b → ∞, we have ρ0 (b) ∼ 2b . Proof The cubic equation (3) can be explicitly solved in terms of trigonometric functions. The solution that takes respectively z± to ∓b is given by 1 w = 2b sin ϕ 3

where

sin ϕ =

3 [c − f (z, α)] 2b3

(see [7]). Since z = z− is mapped into w = b, we get b = 2b sin 13 ϕ and sin ϕ =

3  1 2 c − 2αz− + z− + (log |z− | + i arg z− ) . 3 2b 2

(19)

Let S±k denote the singular points in the w-plane. When w = S±k , we have S±k = 2b sin 13 ϕ±k and  3 1 2 sin ϕ±k = 3 c − 2αz− + z− + [log |z− | + i(arg z− ± 2kπ)] . (20) 2b 2 Subtracting (20) from (19) gives 3kπ sin ϕ±k = 1 ± 3 i, (21) 2b and   3kπ  π 3kπi ϕ±k = arcsin 1 ± 3 i ∼ − ∓ 3 . 2b 2 2b Using the addition formula of the sine function and the Maclaurin expansions, we obtain    1  3kπi  √3  1  3kπi  1 kπi sin . =b− ∓ − ∓ 3 ∓ 3 S±k ∼ 2b cos 2 3 2b 2 3 2b 2b The two singularities nearest to b are clearly  πi and S1 = b − − 2b

 S−1 = b −

πi . 2b

Similarly, z = z+ is mapped into w = −b. The two singularities nearest to −b are   πi πi   and S−1 = −b + − , S1 = −b + 2b 2b thus Lemma 2 is proved. To estimate εp in (12), we split the contour L into L and L , and define  2 1 3 2 (1) p −2p 1 hp (w)eN ( 3 w −b w) dw, εp = (−1) N 2πi L and ε(2) p

p

= (−1) N

−2p

1 2πi

 L

hp (w)eN

2 1 3 ( 3 w −b2 w)

dw.

(22)

(23)

Since the asymptotic nature of the expansion (11) has already been established in (13) for bounded α, we shall be concerned only with the case α → ∞, i.e., b → ∞. Take L ={w ∈ L : |w − b| ≤ cbθ }, where c and θ are constants satisfying c > 0 and − 12 < θ ≤ 1. We choose θ close (2) to − 12 , and fix it. It will be shown that the estimate of εp is exponentially small compared to (1) that of εp .

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Let Γ1 be a closed contour embracing the curve L such that length Γ1 = O(bθ ) and distance(Γ1 , L ) ∼ cb−1/2

as

b → ∞,

(24)

 ¯ 1 be the closure of Ω1 . c < π/2. Let Ω1 denote the domain bounded by Γ1 , and let Ω ¯  be the mirror images of Γ1 and Ω ¯ 1 with respect to the y-axis, and Furthermore, let Γ1 and Ω 1  ¯θ = Ω ¯1 ∪ Ω ¯ 1 . From Lemma 2, we know that h0 (w) is analytic in Ω ¯ θ , so sup |h0 (w)| is put Ω finite. Lemma 3

¯θ w∈Ω

There exists a constant c0 independent of b such that sup |h0 (w)| ≤ c0 |h0 (b)|

as b → ∞.

¯θ w∈Ω

(25)

Proof From (8) we obtain h0 (b) = h0 (−b) =

b1/2 . (α2 − 1)1/4

¯ 1 . It is evident from (8) that we need to estimate To bound h0 (w), first we consider w ∈ Ω z − z− in terms of w − b. From (3), we have 2α(z − z− ) − (z − z− )(z + z− ) −

z − z− 1 1 log(1 + ) = (w + 2b)(w − b)2 . 2 z− 3

(26)

For |w − b| ∼ cbθ with − 12 < θ < 0, the right-hand side of (26) behaves like c2 b2θ+1 . Since 1 z− ∼ 4α by (2), if α(z − z− ) = O(1) then the left-hand side of (26) is bounded, which is a contradiction. Thus, from (26) we get 2α(z − z− ) ∼ c2 b2θ+1

as b → ∞.

(27)

Furthermore, we obtain from (8) and (27)   θ+ 12 √  h0 (w)  2α 1 1  ∼ √cb  ∼ 2.  h0 (b)  θ 2 1/4 1/2 1/2 2α cb (α − 1) b

(28)

¯  , we have By a parallel argument, one can show that when w ∈ Ω 1   θ+1  h0 (w)  1 cb 1 1   2  h0 (b)  ∼ α cbθ+ 12 (α2 − 1)1/4 b1/2 ∼ 1. This together with (28) certainly implies (25). Similar to (24), we define Γ2 to be a closed contour enclosing L such that length Γ2 = O(bθ ) and distance(Γ2 , L ) ∼

1 −1/2 cb 2

as

b → ∞.

¯ 2 be the Note that Γ2 lies inside Γ1 . Let Ω2 denote the domain bounded by Γ2 , and let Ω   ¯ ¯ closure of Ω2 . Also, we let Γ2 and Ω2 be the reflections of Γ2 and Ω2 with respect to the y-axis, ¯ = Ω ¯2 + Ω ¯ 2 and respectively. Put Ω θ ˜ p = sup { |hp (w)| : h

¯ θ }. w∈Ω

(29)

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Proposition 1 Let hp (w) be the sequence of functions defined recursively by (9)-(10). For p = 1, 2, · · ·, there exists a constant Cp > 0, not depending on b, such that and ˜ p ≤ Cp bθ+ 12 |h0 (b)| h

as b → ∞.

(30)

Proof Let ¯ 2 } ∪ {(u, w) ∈ C2 : u ∈ Γ1 , w ∈ Ω ¯ 2 }. Ω(b) = {(u, w) ∈ C2 : u ∈ Γ1 , w ∈ Ω

(31)

Since u ∈ Γ1 , u = b + O(bθ ) as b → ∞. From (16), it follows |Rp (u, w, b)| ≤

kp,i p−1

|Cij |

i=0 j=0

(2b)i−j . |u − w|p+1−i−j |u − b|p+i bp+i

¯ 2 , we have |u − w| ≥ 1 cb−1/2 and |u − b| ≥ 1 cb−1/2 . The argument Also, for u ∈ Γ1 and w ∈ Ω 2 2 ¯  . So we have is the same when u ∈ Γ1 and w ∈ Ω 2 1

sup (u,w)∈Ω(b)

|Rp (u, w, b)| ≤ Ap b 2

as b → ∞,

(32)

where Ap is a constant independent of b. By the Cauchy integral formula, we have hp (w) =

1 2πi

 Γθ

R0 (u, w, b)hp (u)du,

¯ 2 or w ∈ Ω ¯  . By using (9), (10), and where Γθ is either Γ1 or Γ2 depending on whether w ∈ Ω 2 (15), we obtain upon integration by parts  d 1 R0 (u, w, b)ψp−1 (u)du hp (w) = − 2πi Γθ du  1 = (u2 − b2 )R1 (u, w, b)ψp−1 (u)du 2πi Γθ   1 1 = R1 (u, w, b)hp−1 (u)du − R1 (u, w, b)(ap−1 + bp−1 u)du. 2πi Γθ 2πi Γθ

(33)

˜ p−1 O(b−3 ). Since the expression for bp−1 has a factor b−1 , by (17) the last integral in (33) is h This process can be repeated, and we obtain hp (w) =

1 2πi

 Γθ

˜ p−1 O(b−3 ) + · · · + h ˜ 0 O(b−3p ) Rp (u, w, b)h0 (u)du + h

(34)

as b → ∞. By (25) and (32),    1   2πi

Γθ

  1 Rp (u, w, b)h0 (u)du ≤ Ap bθ+ 2 |h0 (b)|,

where Ap is just another constant not depending on b. The desired estimate (30) now follows from (34).

No.4

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Estimation of the Remainder (1)

An estimate for εp can be obtained easily from Proposition 1, and we have 2

1

4

−2p− 3 θ+ 2 b |h0 (b)||Ai(N 3 b2 )| , | ε(1) p | ≤ Cp N

(35)

here and thereafter, Cp is a constant independent of b. (2) To construct an error bound for εp , we define R(w, r) = r|w−1 |,

(36)

where r > 0 does not depend on w and b, and prove the following result. Lemma 4 There is a neighborhood Ω0 (b) of L such that for each w ∈ L a disk with center w and radius R is contained in Ω0 (b) . Proof Recall two singularities of h0 (w) nearest to L are located  from Lemma 2 that the  1 πi π − 12 as b → ∞, and distance(S±1 , L) ∼cb− 2 at S±1 = b − ∓ 2b . Thus, |S±1 − b| ∼ 2b for some constant c. Since distance(S±1 , L ) >distance(S±1 , L), we need only prove R < distance(S±1 , L). By definition, |w − b| ≥ cbθ , when w ∈ L , where − 12 < θ < 0. Hence, |w| > cb, and R = r|w−1 | < cb−1 < distance(S±1 , L), c being a generic symbol for constants whose value may change from place to place. Lemma 5 There exists a constant C0 > 0 such that for w ∈ Ω0 (b) ∪ L, we have |h0 (w)| ≤ C0 |h0 (b)|

as b → ∞.

(37)

Proof As in the proof of Lemma 3, our argument makes heavy use of equation (26). First we note that w can be unbounded. Hence, we divide our discussion into two cases: (i) b = o(w) and (ii) w b. Since L is the unbounded portion of the steepest descent path in Figure 2, there is no w ∈ L satisfying w = o(b), we also note that the case |w − b| = o(1) is already included in case (2). In case (i), the right-hand side of (26) is of order w3 . This implies that the left-hand side of the equation must be of order z 2 , as otherwise one will get a contradiction. Thus, we have   2 1/2 2 3/4  h0 (w)  1 |w| α 3 |w| b b1/4   2 4  h0 (b)  |z| |z|2 b1/2 |w| |w|3 b1/2 = w1/4 = o(1); i.e., (37) holds. In case (ii), the right-hand side of (26) is of order b3 α2 . This implies z α. Then we have   2 1/2 2 1/2  h0 (w)  1 |w| α 1 b α b3/2   2 2  h0 (b)  |z| |z|2 b1/2 α α2 b1/2 = α 1, again proving (37). Now we can use the following result proved in [4, (A.5)]: 4

1

1

 2θ+1

 −2p+1 |ε(2) Ai(N 3 b2 )b−(θ+ 2 ) e− 2 δ b p | ≤ Cp |h0 (b)|N

N2

.

(38) (1)

Since 2θ + 1 > 0, this bound is exponentially small in comparison with that for εp . The following theorem is the main result of this article, and it is a consequence of (35) and (38).

842 Theorem 1 satisfying

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For any integer p ≥ 1, the expansion in (11) holds with the remainder εp 1 2 4 |εp | ≤ C˜p bθ+ 4 N −2p− 3 Ai(N 3 b2 )

as b → ∞,

where C˜p is again some constant independent of b. Now we have extended the validity of expansion (11) to all positive values of α, that is, (11) is indeed an asymptotic expansion which holds uniformly with respect to all α ≥ 0. Acknowledgments I would like to express my deepest gratitude to Professor Roderick S.C. Wong and Professor Chen Hua for the long time guidance, encouragement and invaluable suggestions, and would like to thank my classmate for useful discussions. References 1 Bleistein N. Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm Pure Appl Math, 1966, 19: 353–370 2 Chester C, Friedman B, Ursell F. An extension of the method of steepest descents. Proc Cambridge Philos Soc, 1957, 53: 599–611 3 Erd´ elyi A, Kennedy M, McGregor J L. Parabolic cylinder functions of large order. J Rat Mech Anal, 1954, 3: 459–481 4 Olde Daalhuis A B, Temme N M. Uniform Airy-type expansions of integrals. SIAM J Math Anal, 1994, 25: 304–321 5 Olver F W J. Uniform asymptotic expansions for Weber parabolic cylinder functions of large order. J Res Nat Bur Standards, Sect B, 1959, 63: 131–169 6 Skovgaard H. Asymptotic forms of Hermite polynomials. Technical Report, 18, Department of Mathematics, California Institute of Technology, 1959 7 Sun Weifu, Uniform Asymptotic Expansions of Hermite Polynomials [Thesis (M.Phil.)]. City University of Hong Kong, 1997 8 Szeg˝ o G. Orthogonal Polynomials. Colloquium Publications, Vol 23. 3rd ed. Providence, RI: Amer Math Soc, 1967 9 Wyman M. The asymptotic behaviour of the Hermite polynomials. Canad J Math, 1963, 15: 332–349