Hypergeometric origins of Diophantine properties associated with the Askey scheme

Abstract

The “Diophantine” properties of the zeros of certain polynomials in the Askey scheme, recently discovered by Calogero and his collaborators, are 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 q -analogues of the summation theorems, zeros of these polynomials explicitly which are no longer integer valued. This goes beyond the results obtained by the authors previously mentioned.