Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials

It has been recently pointed out that several orthogonal polynomials of the Askey table admit asymptotic expansions in terms of Hermite and Laguerre polynomials [Lopez and Temme, Meth. Appl. Anal. 6 (1999) 131-146; J. Comp. Appl. Math. 133 (2001) 623-633]. From those expansions, several known and new limits between polynomials of the Askey table were obtained in [Lopez and Temme, Meth. Appl. Anal. 6 (1999) 131-146; J. Comp. Appl. Math. 133 (2001) 623-633]. In this paper, we make an exhaustive analysis of the three lower levels of the Askey scheme which completes the asymptotic analysis performed in [Lopez and Temme, Meth. Appl. Anal. 6 (1999) 131-146; J. Comp. Appl. Math. 133 (2001) 623-633]: (i) We obtain asymptotic expansions of Charlier, Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Hermite polynomials. (ii) We obtain asymptotic expansions of Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Charlier polynomials. (iii) We give new proofs for the known limits between polynomials of these three levels and derive new limits. (C) 2003 Elsevier Inc. All rights reserved.