Hypergeometric Origins of Diophantine Properties Associated With the Askey Scheme

TL;DR

This paper explains the Diophantine properties of zeros in Askey scheme polynomials using hypergeometric series summation theorems, extending the analysis to basic hypergeometric polynomials with explicit zeros.

Contribution

It introduces a hypergeometric series approach to understanding zeros of Askey scheme polynomials and extends the method to basic hypergeometric polynomials with explicit zeros.

Findings

Zeros of certain Askey scheme polynomials are explained via hypergeometric summation theorems.

The method is extended to basic hypergeometric polynomials with explicit, non-integer zeros.

New explicit zeros are derived for polynomials associated with basic hypergeometric series.

Abstract

The "Diophantine" property of the zeros of certain polynomials in the Askey scheme, recently discovered by Calogero and his collaborators, is explained, with suitably chosen parameter values, in terms of the summation theorem of hypergeometric series. Here the Diophantine property refers to integer valued zeros. It turns out that the same procedure can also be applied to polynomials arising from the basic hypergeometric series. We found, with suitably chosen parameters and certain $q-$analogue of the summation theorems, zeros of these polynomials explicitly, which are no longer integer valued. This goes beyond the results obtained by the Authors mentioned above.