A $q$-Chaundy representation for the product of two nonterminating basic hypergeometric series and its symmetric generating functions

We derive double product representations of nonterminating basic hypergeometric series using diagonalization, a method introduced by Theo William Chaundy in 1943. We also present some generating functions that arise from it in the $q$ and $q$-inverse Askey schemes. Using this $q$-Chaundy theorem which expresses a product of two nonterminating basic hypergeometric series as a sum over a terminating basic hypergeometric series, we study generating functions for the symmetric families of orthogonal polynomials in the $q$ and $q$-inverse Askey scheme. By applying the $q$-Chaundy theorem to $q$-exponential generating functions due to Ismail, we are able to derive alternative expansions of these generating functions and from these, new representations for the continuous $q$-Hermite and $q$-inverse Hermite polynomials which are connected by a quadratic transformation for the terminating basic hypergeometric series representations.