The large $n$ behaviour of the hypergeometric polynomial
  $$\FFF{-n}{\sfrac12}{\sfrac12}{\sfrac12-n}{\sfrac12-n}{-1}$$ is considered by
using integral representations of this polynomial. This ${}_3F_2$ polynomial is
associated with the Catalan-Larcombe-French sequence. Several other
representations are mentioned, with references to the literature, and another
asymptotic method is described by using a generating function of the sequence.
The results are similar to those obtained by Clark (2004) who used a binomial
sum for obtaining an asymptotic expansion.Comment: 10 pages, 1 figure. Accepted for publication in {\em Analysis and
  Applications

Temme, Nico M.

English

arXiv

The large n behaviour of the hypergeometric polynomial $3F_2 (_{1over2 - n,1over2-n}^{-n,1over2,1over2};-1) is considered by using integral representations of this polynomial. This3F2 polynomial is associated with the Catalan-Larcombe-French sequence.Several other representations are mentioned, with references to the literature,and another asymptotic method is described by using a generatingfunction of the sequence. The results are similar to those obtained byClark (2004) who used a binomial sum for obtaining an asymptotic expansion

Temme, N.M. (Nico)

CWI's Institutional Repository

C e n t r u m  v o o r  W i s k u n d e  e n  I n f o r m a t i c aMASModelling, Analysis and Simulation Modelling, Analysis and SimulationAsymptotics of a 3F2 polynomial associated with the Catalan-Larcombe-French sequenceN.M. TemmeREPORT MAS-R0607 APRIL 2006Centrum voor Wiskunde en Informatica (CWI) is the national research institute for Mathematics and Computer Science. It is sponsored by the Netherlands Organisation for Scientific Research (NWO).CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics.CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms.Probability, Networks and Algorithms (PNA)Software Engineering (SEN)Modelling, Analysis and Simulation (MAS)Information Systems (INS)Copyright © 2006, Stichting Centrum voor Wiskunde en InformaticaP.O. Box 94079, 1090 GB Amsterdam (NL)Kruislaan 413, 1098 SJ Amsterdam (NL)Telephone +31 20 592 9333Telefax +31 20 592 4199ISSN 1386-3703Asymptotics of a 3F2 polynomial associated with theCatalan-Larcombe-French sequence2000 Mathematics Subject Classification:  41A60, 33C20, 11B83, 33C10Keywords and Phrases: Catalan-Larcombe-French sequence, asymptotic expansion, hypergeometric polynomial,modified Bessel functionNote: This report is accepted for publication in Analysis and Applications.AbstractThe large n behaviour of the hypergeometric polynomial3F2(−n, 12, 1212− n, 12− n; −1)is considered by using integral representations of this polynomial. This3F2 polynomial is associated with the Catalan-Larcombe-French sequence.Several other representations are mentioned, with references to the liter-ature, and another asymptotic method is described by using a generatingfunction of the sequence. The results are similar to those obtained byClark (2004) who used a binomial sum for obtaining an asymptotic ex-pansion.Asymptotics of a 3F2 polynomial associated withthe Catalan-Larcombe-French sequenceNico M. TemmeCWI, P.O. Box 94079, NL-1090 GB Amsterdam, The Netherlandse-mail: Nico.Temme@cwi.nlApril 2, 2006AbstractThe large n behaviour of the hypergeometric polynomial3F2(−n, 12, 1212− n, 12− n; −1)is considered by using integral representations of this polynomial. This3F2 polynomial is associated with the Catalan-Larcombe-French sequence.Several other representations are mentioned, with references to the liter-ature, and another asymptotic method is described by using a generatingfunction of the sequence. The results are similar to those obtained byClark (2004) who used a binomial sum for obtaining an asymptotic ex-pansion.Mathematics Subject Classification 2000: 41A60, 33C20, 11B83, 33C10.Keywords & Phrases: Catalan-Larcombe-French sequence, asymptotic expan-sion, hypergeometric polynomial, modified Bessel function.1 The problemFind the large n asymptotics off(n) = 3F2(−n, 12 , 1212 − n, 12 − n ; −1)(1.1)Peter Larcombe conjectured that limn→∞ f(n) = 2 and Tom Koornwinder gavea proof, based on dominated convergence. See for details of the proof [6], wherealso a different representation of f(n) is considered in the formf(n) = 2n3F2(−n,− 12n, 12 − 12n12 − n, 12 − n; 1). (1.2)1The equivalence of these two forms follows from a quadratic transformation ofthe 3F2−functions as given in [2, Ex. 4(iv), p.97], that is,3F2 a, b, c1 + a− b, 1 + c− c; z =(1− z)−a3F2 12a, 12 + 12a, 1 + a− b − c1 + a− b, 1 + a− c;−4z(1− z)2 .(1.3)with a = −n, b = c = 12, and z = −1. Another form is given by (see [7, Eq.(A2)])f(n) =n!2n(12)n3F2(−n,−n, 121, 12− n ; −1). (1.4)In [4] an asymptotic expansion of 12f(n) has been derived. The asymptoticanalysis is based on the representationPn =1n!∑p+q=n(2np)(2qq)(2p)! (2q)!p! q!. (1.5)By using the relation(2n)! = 22nn!(12)n, n = 0, 1, 2, . . . , (1.6)it is straightforward to verify that (1.5) can be written asPn =24nn!n∑p=0(12)p(12)p(12)n−p(12 )n−pp! (n− p)! . (1.7)By using(a)n−k = (−1)k (a)n(1− a− n)k , (1.8)it follows thatPn =24n(12)n(12)nn!n!n∑p=0(−1)p (−n)p(12)p(12)pp! (12− n)p(12 − n)p, (1.9)that is,Pn =24n(12)n(12)nn!n!3F2(−n, 12 , 1212 − n, 12 − n ; −1), (1.10)which gives the relation with f(n) by using (1.1):Pn =24n(12)n(12)nn!n!f(n) =(2nn)2f(n). (1.11)2The numbers Pn are for n = 0, 1, 2, , . . . known as the elements of the se-quence (A053175) {1, 8, 80, 896, 10816, . . .}, called the Catalan-Larcombe-Frenchsequence, which is originally discussed by Catalan [3]. See the On-Line Ency-clopedia of Integer Sequences http://www.research.att.com/ njas/sequences/ .In this paper we derive a complete asymptotic expansion of the numbersPn by using integral representations of the corresponding 3F2−functions. Ourresults are the same as those obtained by Clark [4], who used the binomial sumin (1.5) without reference to the 3F2−functions.2 TransformationsWe derive an integral representation of the 3F2−function of (1.1) by using severaltransformations for special functions. We start with the beta integralB(x, y) =Γ(x)Γ(y)Γ(x+ y)=∫ 10tx−1(1− t)y−1 dt (2.12)and use it in the form(12)k(12 − n)k=(−1)kn!√piΓ(n+ 12 )∫ 10tk−12 (1− t)n−k− 12 dt, k = 0, 1, . . . , n. (2.13)We substitute this in the representation of the 3F2−function in (1.1)3F2(−n, 12 , 1212 − n, 12 − n; −1)=n∑k=0(−1)k (−n)k(12)k(12)kk! (12− n)k(12 − n)k. (2.14)This gives after performing the k−summationf(n) =n!pi (12 )n∫ 10t−12 (1− t)n− 12 2F1(−n, 1212 − n ;t1− t)dt. (2.15)We substitute t = sin2(θ/2) and obtainf(n) =n!pi (12 )n∫ pi0cos2n(θ/2)2F1(−n, 1212 − n ; tan2(θ/2))dθ. (2.16)We apply a quadratic transformation (see [1, Eq. 15.3.26]) to obtainf(n) =n!pi (12 )n∫ pi02F1(− 12n, 12 − 12n12 − n; sin2 θ)dθ, (2.17)and use the representation of the Legendre polynomialPn(x) =(2n)!2nn!n!xn2F1(− 12n, 12 − 12n12 − n; x−2). (2.18)3A BCDpipipi/2pi/2θψFigure 1: The domain of integration of the integral in (2.21) and subdomains A, B,C and D.This follows from [1, Eq. (22.3.8)] and givesf(n) =2−nn!n!pi (12 )n (12 )n∫ pi0sinn θ Pn(1sin θ)dθ. (2.19)Next, consider (see [8, p. 204])Pn(z) =1pi∫ pi0(z +√z2 − 1 cosψ)ndψ, n = 0, 1, 2, . . . , (2.20)which gives the double integralf(n) =n!n!pi2 (12 )n (12 )n∫ pi0∫ pi0(1 + cos θ cosψ2)ndθ dψ. (2.21)3 Asymptotic analysisThe landscape of the integrand in (2.21) shows peaks at the boundary points(0, 0) and (pi, pi), where it assumes the value 1. Along the interior lines θ = 12piand ψ = 12pi the integrand has the value 2−n. Inside the squares A and C, seeFigure 1, the value of the integrand is between 2−n and 1, in the squares B andD it is between 0 and 2−n. In addition, the contributions from A and C are thesame, and also those from B and D are the same.From an asymptotic point of view it follows that the integral over the fullsquare equals twice the integral over A, with an error that is of order O(2−n),while the total integral is of order O(1), as n is large. Hence, we concentrate on4the integral over A, and write for large values of nf(n) = 22−nn!n!pi2 (12 )n (12 )n[∫ 12pi0∫ 12pi0(1 + cos θ cosψ)n dθ dψ + En], (3.22)where En = O(2−n). Next, we neglect En and put u = sin(θ/2), v = sin(ψ/2),and obtainf(n) ∼ 8n!n!pi2 (12 )n (12 )n∫ 12√20∫ 12√20(1− u2 − v2 + 2u2v2)n du√1− u2dv√1− v2 .(3.23)For the integrals in (3.22) and (3.23) asymptotic expansions can be obtainedby using Laplace’s method for double integrals; see [10, § VIII.10]). In our casea simpler approach is based on neglecting a part of square A by introducingpolar coordinatesu = r cos t, v = r sin t, 0 ≤ r ≤ 12√2, 0 ≤ t ≤ 12pi. (3.24)This gives (again we make an error in the integral that is of order O(2−n))f(n) ∼ 8n!n!pi2 (12 )n (12 )n∫ pi/20∫ 12√20(1− r2 + 2r4 cos2 t sin2 t)nr dr dt√(1 − r2 cos2 t)(1− r2 sin2 t). (3.25)We change r2 into r, and obtainf(n) ∼ 4n!n!pi2 (12 )n (12 )n∫ pi/20∫ 120(1 − r + 2r2 cos2 t sin2 t)n dr dt√(1− r cos2 t)(1− r sin2 t). (3.26)First the standard method for obtaining asymptotic expansions of a Laplace-type integral can be used (for the r−integral). The second step is done byintegrating the coefficients of this expansion with respect to t.For the r−integral we transform the variable of integration by puttingw = − ln (1− r + 2r2 cos2 t sin2 t) . (3.27)This mapping is one-to-one for r ∈ [0, 12], uniformly with respect to t ∈ [0, 12pi],with corresponding w−interval [0, w0], where w0 = w(12 ).We obtainf(n) ∼ 4n!n!pi2 (12 )n (12 )n∫ pi/20∫ w00e−nwF (w, t) dw dt, (3.28)whereF (w, t) =1√(1− r cos2 t)(1 − r sin2 t)drdw. (3.29)54 Asymptotic expansionWe obtain the asymptotic expansion of w−integral in (3.28) by using Watson’slemma (see [10, § I.5]).The function F (w, t) is analytic in a neighborhood of the origin of thew−plane. We expandF (w, t) =∞∑k=0ck(t)wk (4.30)and substitute this expansion in (3.28). Interchanging the order of summa-tion and integration, and replacing the interval of the w−integrals by [0,∞) (astandard procedure in asymptotics) we obtainf(n) ∼ 4n!n!npi2 (12 )n (12 )n∞∑k=0Ckk!nk, n→∞, (4.31)whereCk =∫ pi/20ck(t) dt, k = 0, 1, 2, . . . . (4.32)The coefficients ck(t) can be obtained by the following method. First we needthe inverse of the transformation defined in (3.27). That is, we need coefficientsbk in the expansionr(w) =∞∑k=0bk(t)wk. (4.33)We can find r(w) from (3.27) as a solution of a quadratic equation, with thecondition r(w) ∼ w as w → 0, that is, b0(t) = 1. However, we can alsodifferentiate (3.27) with respect to r and substitute the expansion (4.33), andsolve for the coefficients bk(t). When we have these coefficients we can expandF (w, t) of (3.29) and find ck(t).The first few coefficients ck(t) arec0(t) = 1,c1(t) =12(−1 + 8s2 − 8s4),c2(t) =18(1− 28s2 + 220s4 − 384s6 + 192s8),c3(t) =148(−1 + 92s2 − 1628s4 + 10752s6 − 24576s8 + 23040s10 − 7680s12),c4(t) =1384(1− 280s2 + 10024s4 − 130848s6 + 773904s8 − 2054400s10+2691840s12− 1720320s14 + 430080s16),c5(t) =13840(−1 + 848s2 − 55328s4 + 1259040s6− 13396560s8+73983360s10− 215329920s12 + 349224960s14−319549440s16 + 154828800s18− 30965760s20)(4.34)6where s = sin2 t. For the corresponding Ck we haveC0 =12pi, C1 = 0, C2 =18pi, C3 =18pi, C4 =55384pi, C5 =1164pi. (4.35)As a next step we can replace in (4.31) the ratios n!/(12)n by the asymptoticexpansionn!(12)n=√piΓ(n+ 1)Γ(n+ 12)∼ √pin∞∑k=0γknk, (4.36)whereγ0 = 1, γ1 =18, γ2 =1128, γ3 = − 51024 , γ4 = − 2132768 , γ5 = 399262144 .(4.37)This finally givesf(n) ∼ 2(1 +14n+1732n2+207128n3+148752048n4+3523758192n5+ . . .). (4.38)5 An alternative methodThe numbers Pn were proposed as “Catalan” numbers by an associate of Cata-lan. They appear as coefficients in the series expansion of an elliptic integral ofthe first kindK(k) =∫ 12pi01√1− k2 sin2 tdt, (5.39)which is transformed and written as a power series in k (through an intermediatevariable); this gives a generating function for the sequence {Pn}. For details werefer to [5].In [9] a generating function for the numbers Pn is given in terms of the squareof a modified Bessel function, and we use this approach to obtain an asymptoticexpansion of f(n). See also [7] for details on this generating function.We consider numbers Fn defined as coefficients in the generating function[ew/2 I0(w/2)]2=∞∑n=0Fnwn. (5.40)By considering the relation of the Bessel function with the confluent hypergeo-metric functions (see [1, Eq. 13.6.3]),ezIν(z) =(12z)νΓ(ν + 1)1F1(ν + 122ν + 1; 2z), (5.41)we can write (5.40) in the form (see also [1, Eq. 13.1.27]),[1F1(121; w)]2= e2w[1F1(121; −w)]2=∞∑n=0Fnwn. (5.42)7This gives the representation for Fn:Fn =n∑k=0(12)kk! k!(12)n−k(n− k)! (n− k)! . (5.43)By using (1.8) it follows thatFn =(12)nn!n!n∑k=0(−1)k (−n)k(−n)k(12)k(12− n)kk! k!, (5.44)orFn =(12)nn!n!3F2(−n,−n, 121, 12− n ; −1). (5.45)It follows from (1.4) thatf(n) =n!n!n!2n (12 )n (12 )nFn. (5.46)From (5.40) we obtainFn =12pii∫C[ew/2 I0(w/2)]2wn+1dw =12pii∫Ce2wwn+1h(w) dw, (5.47)whereh(w) =[e−w/2 I0(w/2)]2=[1F1(121; −w)]2, (5.48)and the contour C is a circle around the origin, or any contour that can beobtained from this circle by using Cauchy’s theorem. The main contributioncomes from the saddle point of e2wwn+1 , that is from w = w0 = n/2.In the standard saddle point method (see [10, § II.4]) a quadratic transfor-mation is used to bring the main part of the integrand in the form of a Gaussian.We can obtain the same expansion by just expanding the function h(w) (whichis slowly varying for w > 0) at the saddle point.First we expand (see [1, Eq. 13.4.9])1F1(121; −w)=∞∑k=0ak(w−w0)k, ak =(−1)k(12)kk! k!1F1(12+ k1 + k; −w0)(5.49)and nexth(w) =∞∑k=0Ak(w − w0)k. (5.50)We substitute this expansion in the second integral in (5.47) and obtain theconvergent expansionFn =∞∑k=0AkΦk, Φk =12pii∫Ce2wwn+1(w − w0)k dw. (5.51)8The functions Φk can be evaluated by using the recursion formula (which easilyfollows from integrating by parts)Φk = − 12 (k − 1)(Φk−1 + w0Φk−2), Φ0 =2nn!, Φ1 = 0. (5.52)An asymptotic expansion can be obtained by using a well-known expansionfor ak defined in (5.49). We have (as follows from [1, Eq. 13.5.1])1F1(ac; −x)∼ x−a Γ(c)Γ(c− a)∞∑m=0(a)m(1 + a− c)mm!xm, x→ +∞, (5.53)from which we can obtain expansions for ak and Ak for large values of w0 = n/2.By using these expansions in (5.51) we obtain an expansion for Fn, and finallyfor f(n by using (5.46). This expansion is the same as the one in (4.38).AcknowledgmentI wish to thank Peter Larcombe for suggesting this problem, for encouragingme to investigate the asymptotic properties of the sequence {Pn}, and for in-troducing me to the literature, in particular to the papers [4] – [7].References[1] Milton Abramowitz and Irene A. Stegun. Handbook of mathematical func-tions with formulas, graphs, and mathematical tables, volume 55 of NationalBureau of Standards Applied Mathematics Series. For sale by the Super-intendent of Documents, U.S. Government Printing Office, Washington,D.C., 1964.[2] W. N. Bailey. Generalized hypergeometric series. Cambridge Tracts inMathematics andMathematical Physics, No. 32. Stechert-Hafner, Inc., NewYork, 1964. Reprinting of the first edition published in 1935 by CambridgeUniversity Press.[3] E. Catalan. Nombres de Segner. Rend. Circ. Mat. Pal., 1:190–201, 1887.[4] Lane Clark. An asymptotic expansion for the Catalan-Larcombe-Frenchsequence. J. Integer Seq., 7(2):Article 04.2.1, 5 pp. (electronic), 2004.[5] A. Frazer Jarvis, Peter J. Larcombe, and David R. French. Linear recur-rences between two recent integer sequences. Congr. Numer., 169:79–99,2004.[6] Peter J. Larcombe. Formal proofs of the limiting behaviour of two finiteseries using dominated convergence. Manuscript.9[7] Peter J. Larcombe and David R. French. A new generating function for theCatalan-Larcombe-French sequence: proof of a result by Jovovic. Congr.Numer., 166:161–172, 2004.[8] Nico M. Temme. Special functions. A Wiley-Interscience Publication. JohnWiley & Sons Inc., New York, 1996. An introduction to the classical func-tions of mathematical physics.[9] Nico M. Temme. Large parameter cases of the Gauss hypergeometric func-tion. In Proceedings of the Sixth International Symposium on OrthogonalPolynomials, Special Functions and their Applications (Rome, 2001), vol-ume 153, pages 441–462, 2003.[10] R. Wong. Asymptotic approximations of integrals, volume 34 of Classicsin Applied Mathematics. Society for Industrial and Applied Mathematics(SIAM), Philadelphia, PA, 2001. Corrected reprint of the 1989 original.10

Asymptotics of a ${}_3F_2$ polynomial associated with the Catalan-Larcombe-French sequence

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Asymptotics of a ${}_3F_2$ polynomial associated with the Catalan-Larcombe-French sequence