The power collection method for connection relations: Meixner polynomials. (English) Zbl 1424.33020

Summary: We introduce the power collection method for easily deriving connection relations for certain hypergeometric orthogonal polynomials in the \((q)\)-Askey scheme. We summarize the full-extent to which the power collection method may be used. As an example, we use the power collection method to derive connection and connection-type relations for Meixner and Krawtchouk polynomials. These relations are then used to derive generalizations of generating functions for these orthogonal polynomials. The coefficients of these generalized generating functions are in general, given in terms of multiple hypergeometric functions. From derived generalized generating functions, we deduce corresponding contour integral and infinite series expressions by using orthogonality.


33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
05A15 Exact enumeration problems, generating functions
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
33C20 Generalized hypergeometric series, \({}_pF_q\)


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[1] R. ASKEY, Orthogonal Polynomials and Special Functions, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1975. · Zbl 0298.33008
[2] M. A. BAEDER, H. S. COHL,ANDH. VOLKMER, Generalizations of generating functions for higher continuous hypergeometric orthogonal polynomials in the Askey scheme, Journal of Mathematical Analysis and Applications, 427 (1), 2015. · Zbl 1331.33012
[3] H. CHAGGARA ANDW. KOEPF, Duplication coefficients via generating functions, Complex Variables and Elliptic Equations. An International Journal, 52 (6): 537–549, 2007. · Zbl 1120.33005
[4] H. S. COHL, R. S. COSTAS-SANTOS,ANDP. R. HWANG, Generalizations of generating functions for basic hypergeometric orthogonal polynomials, submitted, 2016.
[5] H. S. COHL, C. MACKENZIE,ANDH. VOLKMER, Generalizations of generating functions for hypergeometric orthogonal polynomials with definite integrals, Journal of Mathematical Analysis and Applications, 407 (2): 211–225, 2013. · Zbl 1306.33016
[6] R. S. COSTAS-SANTOS ANDJ. F. S ´ANCHEZ-LARA, Extensions of discrete classical orthogonal polynomials beyond the orthogonality, Journal of Computational and Applied Mathematics, 225 (2): 440– 451, 2009. · Zbl 1167.42008
[7] F. W. J. OLVER, A. B. OLDEDAALHUIS, D. W. LOZIER, B. I. SCHNEIDER, R. F. BOISVERT, C. W. CLARK, B. R. MILLER ANDB. V. SAUNDERS, eds. NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.13 of 2016-09-16.
[8] M. FOUPOUAGNIGNI, W. KOEPF,ANDD. D. TCHEUTIA, Connection and linearization coefficients of the Askey-Wilson polynomials, Journal of Symbolic Computation, 53: 96–118, 2013. · Zbl 1273.33003
[9] M. FOUPOUAGNIGNI, W. KOEPF, D. D. TCHEUTIA,ANDP. NJIONOUSADJANG, Representations of q -orthogonal polynomials, Journal of Symbolic Computation, 47 (11): 1347–1371, 2012. · Zbl 1247.33015
[10] G. GASPER, Projection formulas for orthogonal polynomials of a discrete variable, Journal of Mathematical Analysis and Applications, 45: 176–198, 1974. · Zbl 0276.33026
[11] M. E. H. ISMAIL, Classical and Quantum Orthogonal Polynomials in One Variable, volume 98 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2005, with two chapters by Walter Van Assche.
[12] M. E. H. ISMAIL ANDM. RAHMAN, Connection relations and expansions, Pacific Journal of Mathematics, 252 (2): 427–446, 2011. · Zbl 1235.05020
[13] M. E. H. ISMAIL ANDP. SIMEONOV, Formulas and identities involving the Askey-Wilson operator, Advances in Applied Mathematics, 76: 68–96, 2016. · Zbl 1336.33026
[14] M. E. H. ISMAIL ANDD. STANTON, Expansions in the Askey-Wilson polynomials, Journal of Mathematical Analysis and Applications, 424 (1):664–674, 2015. · Zbl 1309.33020
[15] R. KOEKOEK, P. A. LESKY,ANDR. F. SWARTTOUW, Hypergeometric orthogonal polynomials and their q -analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010, with a foreword by Tom H. Koornwinder. · Zbl 1200.33012
[16] W. KOEPF ANDD. SCHMERSAU, Representations of orthogonal polynomials, Journal of Computational and Applied Mathematics, 90 (1): 57–94, 1998. · Zbl 0907.65017
[17] E. D. RAINVILLE, Special Functions, The Macmillan Co., New York, 1960.
[18] H. M. SRIVASTAVA ANDPERW. KARLSSON, Multiple Gaussian hypergeometric series, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1985. · Zbl 0552.33001
[19] D. D. TCHEUTIA, On Connection, Linearization and Duplication Coefficients of Classical Orthogonal Polynomials, PhD thesis, Universit¨at Kassel, Kassel, Germany, 2014, viii+145 pages.
[20] D. D. TCHEUTIA, M. FOUPOUAGNIGNI, W. KOEPF,ANDP. NJIONOUSADJANG, Coefficients of multiplication formulas for classical orthogonal polynomials, Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan, 39 (3): 497–531, 2016. · Zbl 1334.33028
[21] N. M. TEMME, Special Functions: an Introduction to the Classical Functions of Mathematical Physics, J. Wiley & Sons, New York, 1996. · Zbl 0856.33001
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