Asymptotics of the associated Pollaczek polynomials

Luo, Min-Jie; Wong, R.

doi:10.1090/proc/14405

Asymptotics of the associated Pollaczek polynomials
HTML articles powered by AMS MathViewer

by Min-Jie Luo and R. Wong PDF

Proc. Amer. Math. Soc. 147 (2019), 2583-2597 Request permission

Abstract:

In this note, we investigate the large-$n$ behavior of the associated Pollaczek polynomials $P_{n}^{\lambda }\left (z;a,b,c\right )$. These polynomials involve four real parameters $\lambda$, $a$, $b$, and $c$, in addition to the complex variable $z$. Asymptotic formulas are derived for these polynomials, when $z$ lies in the complex plane bounded away from the interval of orthogonality $\left (-1,1\right )$, as well as in the interior of the interval of orthogonality. In the process of studying the asymptotic behavior of these polynomials when $z\in \mathbb {C}\setminus [-1,1]$, we found that the existing representations of $P_{n}^{\lambda }\left (z;a,b,c\right )$ do not provide useful information about their large-$n$ asymptotics. Here, we present a new representation in terms of the Gauss hypergeometric functions, from which the large-$n$ asymptotics for $z$ in $\mathbb {C}\setminus [-1,1]$ can be readily obtained. The asymptotic approximation in the interior of the interval of orthogonality is obtained by using asymptotic theory for difference equations.

References

A. D. Alhaidari, H. Bahlouli, and M. E. H. Ismail, The Dirac-Coulomb problem: a mathematical revisit, J. Phys. A 45 (2012), no. 36, 365204, 9. MR 2967907, DOI 10.1088/1751-8113/45/36/365204
George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Rui Bo and R. Wong, Asymptotic behavior of the Pollaczek polynomials and their zeros, Stud. Appl. Math. 96 (1996), no. 3, 307–338. MR 1378865, DOI 10.1002/sapm1996963307
Joaquin Bustoz and Mourad E. H. Ismail, The associated ultraspherical polynomials and their $q$-analogues, Canadian J. Math. 34 (1982), no. 3, 718–736. MR 663314, DOI 10.4153/CJM-1982-049-6
Jairo A. Charris and Mourad E. H. Ismail, On sieved orthogonal polynomials. V. Sieved Pollaczek polynomials, SIAM J. Math. Anal. 18 (1987), no. 4, 1177–1218. MR 892496, DOI 10.1137/0518086
Nadhla A. Al-Salam and Mourad E. H. Ismail, On sieved orthogonal polynomials. VIII. Sieved associated Pollaczek polynomials, J. Approx. Theory 68 (1992), no. 3, 306–321. MR 1152221, DOI 10.1016/0021-9045(92)90107-Y
Jairo A. Charris, Mourad E. H. Ismail, and Sergio Monsalve, On sieved orthogonal polynomials. X. General blocks of recurrence relations, Pacific J. Math. 163 (1994), no. 2, 237–267. MR 1262296
E. Hendriksen and H. van Rossum, Orthogonal Laurent polynomials, Nederl. Akad. Wetensch. Indag. Math. 48 (1986), no. 1, 17–36. MR 834317
Erik Hendriksen, Associated Jacobi-Laurent polynomials, J. Comput. Appl. Math. 32 (1990), no. 1-2, 125–141. Extrapolation and rational approximation (Luminy, 1989). MR 1091783, DOI 10.1016/0377-0427(90)90424-X
Erik Hendriksen, A weight function for the associated Jacobi-Laurent polynomials, J. Comput. Appl. Math. 33 (1990), no. 2, 171–180. MR 1090894, DOI 10.1016/0377-0427(90)90367-9
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2009. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey; Reprint of the 2005 original. MR 2542683
Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
Henri Padé, Recherches sur la convergence des développements en fractions continues d’une certaine catégorie de fonctions, Ann. Sci. École Norm. Sup. (3) 24 (1907), 341–400 (French). MR 1509084
Félix Pollaczek, Systèmes de polynomes biorthogonaux qui généralisent les polynomes ultrasphériques, C. R. Acad. Sci. Paris 228 (1949), 1998–2000 (French). MR 31126
Félix Pollaczek, Sur une famille de polynômes orthogonaux à quatre paramètres, C. R. Acad. Sci. Paris 230 (1950), 2254–2256 (French). MR 35868
Félix Pollaczek, Sur une généralisation des polynomes de Jacobi, Mémor. Sci. Math., no. 131, Gauthier-Villars, Paris, 1956 (French). MR 0075339
G. Szegö, On certain special sets of orthogonal polynomials, Proc. Amer. Math. Soc. 1 (1950), 731–737. MR 42546, DOI 10.1090/S0002-9939-1950-0042546-2
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
X.-S. Wang and R. Wong, Asymptotics of orthogonal polynomials via recurrence relations, Anal. Appl. (Singap.) 10 (2012), no. 2, 215–235. MR 2903981, DOI 10.1142/S0219530512500108
Jet Wimp, Explicit formulas for the associated Jacobi polynomials and some applications, Canad. J. Math. 39 (1987), no. 4, 983–1000. MR 915027, DOI 10.4153/CJM-1987-050-4
Jet Wimp, Pollaczek polynomials and Padé approximants: some closed-form expressions, J. Comput. Appl. Math. 32 (1990), no. 1-2, 301–310. Extrapolation and rational approximation (Luminy, 1989). MR 1091799, DOI 10.1016/0377-0427(90)90440-B
R. Wong and H. Li, Asymptotic expansions for second-order linear difference equations, J. Comput. Appl. Math. 41 (1992), no. 1-2, 65–94. Asymptotic methods in analysis and combinatorics. MR 1181710, DOI 10.1016/0377-0427(92)90239-T