The purpose of this article is to bring structure to (basic) hypergeometric
biorthogonal systems, in particular to the q-Askey scheme of basic
hypergeometric orthogonal polynomials. We aim to achieve this by looking at the
limits as p->0 of the elliptic hypergeometric biorthogonal functions from
Spiridonov, with parameters which depend in varying ways on p. As a result we
get 38 systems of biorthogonal functions with for each system at least one
explicit measure for the bilinear form. Amongst these we indeed recover the
q-Askey scheme. Each system consists of (basic hypergeometric) rational
functions or polynomials.Comment: 27 pages. This is a self-contained article which can also be seen as
  part 1 of a 3 part series on limits of (multivariate) elliptic hypergeometric
  biorthogonal functions and their measure

Askey

Atakishiyev

Eric M. Rains

Fokko J. van de Bult

Gasper

Koekoek

Macdonald

Pastro

Rains

Rosengren

Spiridonov

van de Bult

English

arXiv

In this article we extend the results of our article "Limits of elliptic hypergeometric biorthogonal functions" to the multivariate setting. In that article we determined which families of biorthogonal functions arise as limits from the elliptic hypergeometric biorthogonal functions from Spiridonov when p → 0. Here we show that the classification of the possible limits of the BC_n type multivariate biorthogonal functions previously introduced by the second author is identical to the univariate classification. That is, for each univariate limit family there exists a multivariate extension, and in particular we obtain multivariate versions for all elements of the q-Askey scheme. For the Askey-Wilson polynomials these are the Koornwinder polynomials, and the multivariate versions of the Pastro polynomials form a two-parameter family which include the Macdonald polynomials

Limits of elliptic hypergeometric biorthogonal functions

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