Some generating functions for the Laguerre and related polynomials Poh-Aun Lee a,1 , Seng-Huat Ong b,2 , H.M. Srivastava c, * a Faculty of Information Technology, University of Telekom, Jalan Ayer Keroh Lama, 75450 Melaka, Malaysia b Department of Mathematics, University of Malaya, 50603 Kuala Lumpur, Malaysia c Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada V8W 3P4 Abstract The authors investigate (and present generalizations of) several generating functions of the classical Laguerre and related polynomials. One of these generating functions was recently rederived probabilistically by Lee [P.-A. Lee, Bull. Inst. Math. Acad. Sinica 25 (1997) 151±154], who made use of the theory of a class of probability distributions known as the non-central negative binomial distribution. Basic (or q-) extensions of the main series identities are also considered. Ó 2000 Elsevier Science Inc. All rights re- served. AMS: 33C20; 33C45; 33D15; 33D20; 11B65; 60E05 Keywords: Laguerre polynomials; Binomial distribution; Bessel polynomials; Jacobi polynomials; Hypergeometric and basic (or q-) hypergeometric functions; Kummer's transformation; Binomial and q-binomial expansions; Heine's (or q-Euler) transformations; q-series identities 1. Introduction, de®nitions, and preliminaries The classical Laguerre polynomials L a n x, of order a and degree n in x, de®ned by Applied Mathematics and Computation 108 (2000) 129±138 www.elsevier.nl/locate/amc * Corresponding author. E-mail: hmsri@uvvm.uvic.ca 1 E-mail: palee@unitele.com.my 2 E-mail: ong@omega.math.um.edu.my 0096-3003/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII:S0096-3003(99)00007-7