Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series

We give a method to embed the q-series in a (p, q)-series and derive the corresponding (p, q)-extensions of the known q-identities. The (p, q)-hypergeometric series, or twin-basic hypergeometric series (different from the usual bibasic hypergeometric series), is based on the concept of twin-basic number [n]p,q = (p n−qn)/(p−q). This twin-basic number occurs in the theory of two-parameter quantum algebras and has also been introduced independently in combinatorics. The (p, q)-identities thus derived, with doubling of the number of parameters, offer more choices for manipulations; for example, results that can be obtained via the limiting process of confluence in the usual q-series framework can be obtained by simpler substitutions. The q-results are of course special cases of the (p, q)-results corresponding to choosing p = 1. This also provides a new look for the q-identities.