Centrum voor Wiskunde en Informatica REPORTRAPPORT The Role of Hermite Polynomials in Asymptotic Analysis N.M. Temme, J.L. López Modelling, Analysis and Simulation (MAS) MAS-R9926 September 30, 1999 Report MAS-R9926 ISSN 1386-3703 CWI P.O. Box 94079 1090 GB Amsterdam The Netherlands CWI is the National Research Institute for Mathematics and Computer Science. CWI is part of the Stichting Mathematisch Centrum (SMC), the Dutch foundation for promotion of mathematics and computer science and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of ERCIM, the European Research Consortium for Informatics and Mathematics. Copyright © Stichting Mathematisch Centrum P.O. Box 94079, 1090 GB Amsterdam (NL) Kruislaan 413, 1098 SJ Amsterdam (NL) Telephone +31 20 592 9333 Telefax +31 20 592 4199 1 The Role of Hermite Polynomials in Asymptotic Analysis Nico M. Temme1 and José L. López2 1 2 CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands Departamento de Matématica Aplicada, Facultad de Ciencias, Universidad de Zaragoza, 50013-Zaragoza, Spain e-mail: nicot@cwi.nl, jllopez@posta.unizar.es ABSTRACT Hermite polynomials are considered as approximants in asymptotic representations of certain other polynomials. Examples are given for polynomials from the Askey scheme of hypergeometric orthogonal polynomials. We also mention that Hermite polynomials can be used as main approximants in uniform asymptotic representations of certain types of integrals and differential equations. 1991 Mathematics Subject Classification: 33C45, 41A60, 41A10. Keywords & Phrases: Limits of polynomials, Hermite polynomials, Laguerre polynomials, orthogonal polynomials, Askey scheme, asymptotic expansions. Note: Work carried out under project MAS1.3 Partial differential equations in porous media research. This report has been accepted for publication in the proceedings of the International Workshop on Special Functions, Hong Kong, June 21 - 25, 1999. 1. Introduction Hermite polynomials show up in several problems of asymptotic analysis. We consider three different instances where these classical orthogonal polynomials can be used as main approximants: 1. 2. 3. As limits of other polynomials such as Laguerre and Jacobi orthogonal polynomials, but also of generalized Bernoulli polynomials. We explain that these limits may be derived from asymptotic representations in which Hermite polynomials occur. In turning point problems for second order linear differential equations. In particular the Hermite polynomials can be used when two nearby turning points are present. For functions defined as an integral in which the saddle points follow a certain pattern for certain values of the parameters. 2 In all three cases, asymptotic representations of polynomials are considered in terms of Hermite polynomials. In the second and third case the Hermite polynomials arise as special cases of another set of special functions, the Weber parabolic cylinder functions, which can be used in similar and more general problems of asymptotic analysis. We concentrate on the first topic; in particular we give examples from the Askey scheme of hypergeometric polynomials. We give examples in which Hermite polynomials are used in asymptotic approximations, and we also give approximations in terms of other polynomials. 2. Limits between orthogonal polynomials It is well known that the Hermite polynomials play a crucial role in certain limits of the classical orthogonal polynomials. For example, the ultraspherical (Gegenbauer) polynomials Cnγ (x), which are defined by the generating function (1 − 2xw + w2 )−γ = ∞ X Cnγ (x)wn , n=0 −1 ≤ x ≤ 1, |w| < 1, have the well-known limits Cnγ (x) = xn , γ γ→∞ Cn (1) lim 1 √ lim γ −n/2 Cnγ (x/ γ ) = Hn (x). n! γ→∞ The first limit shows that the zeros of Cnγ (x) tend to the origin if the order γ tends to infinity. The second limit is more interesting; it gives the relation with the Hermite polynomials if the order becomes large and the argument x is properly scaled. For the Laguerre polynomials, which are defined by the generating function (1 − w)−α−1 e−wx/(1−w) = ∞ X n=0 n Lα n (x) w , |w| < 1, (2.1) α, x ∈ C, similar results are  √ 
(−1)n 2−n/2 √ lim α−n/2 Lα Hn x/ 2 . n x α +α = α→∞ α→∞ n! (2.2) This again gives insight in the location of the zeros for large values of the order α, and the relation with the Hermite polynomials if the order becomes large and x is properly scaled. lim α−n Lα n (αx) = (1 − x)n , n! Many methods are available to prove these and other limits. In this paper we concentrate on asymptotic relations between the polynomials, from which the limits follow as special cases. 3. The Askey scheme In Koekoek and Swarttouw (1998) many relations are given for hypergeometric orthogonal polynomials and their q−analogues, including limit relations between many polynomials. In Figure 1 we show examples for which limit relations between neighboring polynomials 3 are available, but many other limit relations are mentioned in Koekoek and Swarttouw (1998), Godoy et al. (1998) and Ronveaux et al. (1998) In López & Temme (1999a, 1999b) we have given several asymptotic relations between polynomials and Hermite polynomials. In the first paper we considered Gegenbauer polynomials, Laguerre polynomials, Jacobi polynomials and Tricomi-Carlitz polynomials. In the second paper we have considered generalized Bernoulli polynomials, generalized Euler polynomials, generalized Bessel polynomials and Buchholz polynomials. The method for all these cases is the same and we observe that the method also works for polynomials outside the class of hypergeometric polynomials. The method is different from the one described in Godoy et al. (1998), where also more terms in the limit relation are constructed in order to obtain more insight in the limiting process. Askey Scheme of Hypergeometric Orthogonal Polynomials 4 F3 Wilson Racah n, x, a, b, c, d n, x, α, β, γ, δ Continuous 3 F2 dual Hahn 2 F1 Continuous Hahn Hahn n, x, a, b, c n, x, a, b, c n, x, α, β, N Meixner Pollaczek Jacobi Meixner n, x, α, β n, x, β, c n, x, φ, λ Laguerre 1 F1 2 F0 n, x, α 2 F0 Dual Hahn n, x, γ, δ, N Krawtchouk n, x, p, N Charlier n, x, a Hermite n, x Figure 1. The Askey scheme for hypergeometric orthogonal polynomials, with indicated limit relations between the polynomials. 4 In current research we investigate if other limits in the Askey scheme can be replaced by asymptotic results. Until now we verified all limits from the third level to the fourth (Laguerre and Charlier) and the fifth level (Hermite). Several limits are new, and all results have full asymptotic expansions. 4. Asymptotic representations Starting point is a generating series ∞ X F (x, w) = pn (x) wn , (4.1) n=0 F is a given function, which is analytic with respect to w at w = 0, and pn is independent of w. The relation (4.1) gives for pn the Cauchy-type integral Z 1 dw pn (x) = F (x, w) n+1 , 2πi C w where C is a circle around the origin inside the domain where F is analytic (as a function of w). We write 2 F (x, w) = eAw−Bw f (x, w), where A and B do not depend on w. This gives Z 2 dw 1 eAw−Bw f (x, w) n+1 . pn (x) = 2πi C w (4.2) Because f is also analytic (as a function of w), we can expand −Aw+Bw 2 f (x, w) = e F (x, w) = ∞ X ck wk , (4.3) k=0 that is, 1 2 f (x, w) = 1 + [p1 (x) − A]w + [p2 (x) − Ap1 (x) + B + A2 ]w2 + . . . if we assume that p0 (x) = 1 (which implies c0 = 1). We substitute (4.3) in (4.2). The Hermite polynomials have the generating function 2xw−w 2 e ∞ X Hn (x) n = w , n! n=0 x, w ∈ C, which gives the Cauchy-type integral n! Hn (x) = 2πi Z C 2 e2xz−z z −n−1 dz, (4.4) 5 where C is a circle around the origin and the integration is in positive direction. The result is the finite expansion n X ck Hn−k (ξ) pn (x) = z , z k (n − k)! n z= √ B, k=0 A ξ= √ , 2 B (4.5) because terms with k > n do not contribute in the integral in (4.2). In order to obtain an asymptotic property of (4.5) we take A and B such that c1 = c2 = 0. This happens if we take 1 A = p1 (x), B = p21 (x) − p2 (x). 2 As we will show, the asymptotic property follows from the behavior of the coefficients ck if we take a parameter of the polynomial pk (x) large. We use the following lemma, and explain what happens by considering a few examples. Lemma 4.1. Let φ(w) be analytic at w = 0, with Maclaurin expansion of the form φ(w) = µwn (a0 + a1 w + a2 w2 + . . .), where n is a positive integer and ak are complex numbers that do not depend on the complex number µ, a0 6= 0. Let ck denote the coefficients of the power series of f (w) = eφ(w) , that is, ∞ X ck wk . f (w) = eφ(w) = k=0 Then c0 = 1, ck = 0, k = 1, 2 . . . , n − 1 and   ck = O |µ|⌊k/n⌋ , µ → ∞. Proof. The proof follows from expanding ∞ X k φ(w) ck w = e = ∞ X [φ(w)]k k=0 k=0 = ∞ X µk wkn k=0 k! k! (a0 + a1 w + a2 w2 + . . .)k , and comparing equal powers of w. 4.1. Ultraspherical polynomials The generating function is 2 −γ F (x, w) = (1 − 2xw + w ) = ∞ X n=0 Cnγ (x) wn , 6 with C0γ (x) = 1, C1γ (x) = 2γx, C2γ (x) = 2γ(γ + 1)x2 − γ. Hence, A = C1γ (x) = 2xγ, B= and we can write Cnγ (x) 1 2 2 [C1γ (x)] − C2γ (x) = γ(1 − 2x2 ), n X ck Hn−k (ξ) =z , z k (n − k)! n (4.6) k=0 where z= We have c0 = 1, p γ(1 − 2x2 ) , c1 = c2 = 0, ξ= xγ . z c3 = 2 γx(4x2 − 3). 3 Higher coefficients follow from a recursion relation. The function f (x, w) of (4.3) has the form f (x, w) = eφ(x,w) , where φ(x, w) = γw3 (a0 + √ a1 w + a2 w2 + . . .). By using Lemma 4.1 and ξ = O( γ ) we conclude that the sequence {φk } with φk = ck /z k Hn−k (ξ) has the following asymptotic structure:   φk = O γ n/2+⌊k/3⌋−k , k = 0, 1, 2, . . . . This explains the asymptotic nature of the representation in (4.6) for large values of γ, with x and n fixed. To verify the limits given in (1.2) and (1.3), we first write x in terms of ξ: ξ x= p . γ + 2ξ 2 With this value of x we can verify that ck /z k = o(1), γ → ∞, k > 0, and in fact we have the limit γn lim Cnγ γ→∞ (γ + 2x2 )n/2 x p γ + 2x2 ! = 1 Hn (x). n! 4.2. Laguerre polynomials We take as generating function (see (1.5)) −α−1 wx/(1+w) F (x, w) = (1 + w) e = ∞ X n (−1)n Lα n (x) w . n=0 7 We have Lα 0 (x) = 1, 1 2 Lα 1 (x) = α + 1 − x, 2 Lα 2 (x) = [(α + 1)(α + 2) − 2(α + 2)x + x ], which gives 1 2 A = x − α − 1, B = x − (α + 1), and we obtain n n Lα n (x) = (−1) z n X ck Hn−k (ξ) , z k (n − k)! (4.7) k=0 where r 1 z = x − (α + 1) , 2 ξ= x−α−1 . 2z (4.8) We have c0 = 1, c1 = c2 = 0, 1 3 c3 = (3x − α − 1). Higher coefficients follow from a recursion relation. The representation in (4.7) has an asymptotic character for large values of |α| + |x|. It is not difficult to verify that the limits given in (1.6) and (1.7) follow from (4.7). 5. Meixner-Pollaczek polynomials into Laguerre polynomials We give an example on how to use Laguerre polynomials for approximating other polynomials. Lemma 5.1. Let the polynomials pn (x) be defined by the generating function F (x, w) = ∞ X pn (x) wn , n=0 where F (x, w) is analytic in w = 0 and F (x, 0) = 1. Let the coefficients ck (x) be defined by the expansion e−Aw/(Bw−1) (1 − Bw)C+1 F (x, w) = ∞ X ck (x)wk , c0 = 1, k=0 where A, B and C do not depend on w. Then pn (x) can be represented as the finite sum pn (x) = B n/2 n X ck (x) k=0 B k/2 (C) Ln−k (ξ), ξ= A , B where Lα n (x) are the Laguerre polynomials. Moreover, A, B and C can be chosen such that c1 = 0, c2 = 0, c3 = 0. Proof. Use the Cauchy integral of pn (x) and of the Laguerre polynomials. 8 For the Meixner-Pollaczek polynomials we have the generating function: F (w) = 1 − eiφ w
−λ+ix 1 − e−iφ w
−λ−ix = ∞ X Pn(λ) (x; φ)wn . n=0 From (2.1) it follows that G(w) = eAw/(Bw−1) (1 − Bw)−C−1 = ∞ X n=0 where ξ = A/B. We define ck by f (w) = F (w)/G(w) = for the Meixner-Pollaczek polynomials reads Pn(λ) (x; φ) = n X n n L(C) n (ξ)B w , P∞ (C) B n−k ck Ln−k (ξ), k=0 ck w k . Then the expansion ξ = A/B. k=0 We write x + iλ = reiθ , θ ∈ [0, π], r ≥ 0, and consider r → ∞; the asymptotic results hold uniformly with respect to θ. First we consider a simple case by taking B = 1 and C = α, and solve c1 = 0 for A. This gives A = α + 1 − 2λ cos φ − 2x sin φ. The first coefficients ck are given by c0 = 1, c1 = 0, 1 c2 = x sin 2φ + λ cos 2φ − 2(x sin φ + λ cos φ) + α. 2 The first term approximation can be written as h
i n−1 , Pn(λ) (x; φ) = L(α) n (ξ) + O r ξ = A. In this case a limit can be obtained by putting λ = (α + 1)/2. Then we have ck = O(φ2 ) as φ → 0 for k ≥ 2, and we obtain lim Pn(α+1)/2 [(α + 1)(1 − cos φ) − ξ)/(2 sin φ); φ] = L(α) n (ξ). φ→0 This includes the limit of the Askey scheme lim Pn(α+1)/2 (−ξ/(2φ); φ) = L(α) n (ξ). φ→0 Next we solve c1 = 0, c2 = 0 for A and C, with B = 1. This gives A = 2[x(sin φ − sin 2φ) + λ(cos φ − cos 2φ)], C = 2[x(2 sin φ − sin 2φ) + λ(2 cos φ − cos 2φ)] − 1. 9 and the first term approximation can be written as h
i n−2 Pn(λ) (x; φ) = L(α) (ξ) + O r , n ξ = A, α = C. as r → ∞, uniformly with respect to θ. Solving A = ξ, C = α for x and λ, we obtain 1 λ = (1 − cos φ)ξ + (α + 1)(2 cos φ − 1), 2 2(ξ − α − 1) cos2 φ + (α + 1 − 2ξ) cos φ + α + 1 − ξ x= . 2 sin φ Then c3 = 23 (α + 1 − 2ξ)(1 − cos φ) and ck = O(φ2 ) as φ → 0 for k ≥ 3. We obtain the limit lim Pn(λ) (x; φ) = L(α) n (ξ). φ→0 6. Methods based on differential equations The function 1 2 Un (x) = e− 2 x Hn (x) satisfies the differential equation U ′′ = p(x)U, p(x) = x2 − (2n + 1). (6.1) √ The function p(x) has two real zeros ± 2n + 1 , and all n zeros of Hn (x) are in the interval √ √ (− 2n + 1 , 2n + 1 ). √ Scaling parameters, we see that Vn (t) := Un ( 2n + 1 t) satisfies
d2 V = (2n + 1)2 t2 − 1 V. 2 dt The question now is, can we approximate the solutions of the equation
d2 W = ν 2 t2 − α2 W + f (t)W, 2 dt ν → ∞, in terms of Hermite polynomials or related functions? For more details on this method we refer to Olver (1980) and Temme (1990). 7. Hermite-type approximations for integrals The Hermite polynomials can be represented in the form √
1 2n/2 Hn ξ 2n + 1 = n! (n + 12 )n/2 2πi Z 1 dt e(n+ 2 )ψ(t) √ , t C (7.1) 10 where 1 ψ(t) = 2ξt − ln t − t2 . 2 and the contour C runs from t = −∞, ph t = −π, encircles the origin in positive direction, and terminates at −∞, now with ph t = +π. The saddle points of the integral are defined by the equation ψ′ (t) = 2ξ − 1/t − t = 0 and are given by p t1,2 = ξ ± ξ 2 − 1 . (7.2) When ξ = ±1 the saddle points coalesce, and when ξ ∼ 1 uniform Airy-type expansions can be derived. When −1 < ξ < 1 the saddle points are complex (on the unit circle); for √ √ these values of ξ, that is, if − 2n + 1 < x < 2n + 1 , zeros occur. When ξ > 1 or ξ < −1 the saddle points are real, and the Hermite polynomials are non-oscillating. See Figure 2 for the location of the saddle points. locus of t 2 −1 1 locus of t 1 Figure 2. The location of the two saddle points t1,2 defined in (7.2). 7.1. An expansion in terms of Hermite polynomials We consider integrals of the form 1 Fκ (ξ) = 2πi Z eκΨ(t) f (t) C dt , t (7.3) where 1 2 Ψ(t) = 2ξt − ρ2 ln t − t2 . (7.4) We assume that κ is a positive large parameter and that ρ is positive. The contour C is as in (7.1). The saddle points t1,2 are now given by t1,2 = ξ ± p ξ 2 − ρ2 . (7.5) For large values of κ the function Fκ (ξ) defined in (7.3) can be expanded in terms of parabolic cylinder functions. This asymptotic expansion holds uniformly with respect to 11 ξ ∈ IR and ρ ∈ [0, ∞). For certain values of κ and ρ the parabolic cylinder functions reduce to Hermite polynomials. For more details we refer to Temme (1986), Bo Rui & Wong (1994), Jin & Wong (1996), and Li and Wong (1999). 8. Bibliography [ 1] Bo Rui & R. Wong (1994). Uniform asymptotic expansion of Charlier polynomials, Methods Appl. Anal. 1, 294–313. [ 2] X.-S. Jin & R. Wong (1997). Uniform asymptotic expansions for Meixner polynomials. Manuscript. Constructive Approximation 33, 119–127. [ 3] R. Koekoek & R.F. Swarttouw (1998). The Askey-scheme of hypergeometric orthogonal polynomials and its q−analogue. Technical University Delft. Report 98– 17. [ 4] X. Li & R.Wong (1999). On the asymptotics of the Meixner-Pollaczek polynomials and their zeros. Submitted. [ 5] J.L. López & N.M. Temme (1999a). Approximations of orthogonal polynomials in terms of Hermite polynomials. CWI Report MAS-R9901. Accepted for publication in Methods and Applications of Analysis. [ 6] J.L. López & N.M. Temme (1999b). Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel and Buchholz polynomials. Submitted. [ 7] F.W.J. Olver (1980), Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions, Proc. Roy. Soc. Edinburgh, 86A, 213–234. [ 8] E. Godoy, A. Ronveaux, A. Zarzo and I. Area (1998). On the limit relations between classical continuous and discrete orthogonal polynomials, J. Comp. Appl. Math., 91, 97–105. [ 9] A. Ronveaux, A. Zarzo, I. Area and E. Godoy (1998). Transverse limits in the Askey tableau, J. Comp. Appl. Math., 98, 327–335. [ 10] Temme, N.M. (1986). Laguerre polynomials: Asymptotics for large degree, CWI Report AM–R8610, Amsterdam. [ 11] N.M. Temme (1990). Polynomial asymptotic estimates of Gegenbauer, Laguerre, and Jacobi polynomials, 455–476 in Asymptotic and computational analysis, Proceedings, R. Wong (ed.), Lect. Notes Pure Appl. Math., 124, Marcel Dekker, New York. [ 12] N.M. Temme (1996), Special functions: An introduction to the classical functions of mathematical physics, Wiley, New York. ISBN 0-471-11313-1. [ 13] R. Wong (1989), Asymptotic approximations of integrals, Academic Press, New York. ISBN 0-12-762535-6.