Asymptotic Reductions Between the Wilson Polynomials and the Lower Level Polynomials of the Askey Scheme

In López and Temme (Meth. Appl. Anal. 6, 131–146 (1999); J. Math. Anal. Appl. 239, 457–477 (1999); J. Comp. Appl. Math. 133, 623–633 (2001)), the authors introduced a new technique to analyse asymptotic relations in the Askey scheme. They obtained asymptotic and, at the same time, finite exact representations of orthogonal polynomials of the Askey tableau in terms of Hermite and Laguerre polynomials. That analysis is continued in Ferreira et al. (Adv. Appl. Math. 31(1), 61–85 (2003); Acta Appl. Math. 103(3), 235–252 (2008); J. Comput. Appl. Math. 217(1), 88–109 (2008)), where the authors derived new finite and asymptotic relations between polynomials located in the four lower levels of the Askey tableau. In this paper we complete that analysis obtaining finite exact representations of the Wilson polynomials in terms of the hypergeometric polynomials of the other four levels of the Askey scheme. Using an asymptotic principle based on the “matching” of their generating functions, we prove that these representations have an asymptotic character for large values of certain parameters and provide information on the zero distribution of the Wilson polynomials. A new limit of the Wilson polynomials in terms of Hermite polynomials is obtained as a consequence. Some numerical experiments illustrating the accuracy of the approximations are given.