Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials

  • Authors:

  • Chelo Ferreira;José L. Lopez;Esmeralda Mainar

  • Affiliations:

  • Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Zaragoza, 50013 Zaragoza, Spain;Departamento de Matemática e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain;Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avda. de los Castros s/n, 39005 Santander, Spain

  • Venue:
  • Advances in Applied Mathematics
  • Year:
  • 2003

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Abstract

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.