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

R. Jagannathan·K. Srinivasa Rao
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Abstract

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 (diferent from the usual bibasic hypergeometric series), is based on the concept of twin-basic number [n]_{p,q} = (p^n - q^n)/(p-q). This twin-basic number occurs in the theory of two-parameter quantum algebras and has 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.

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R. Jagannathan, K. Srinivasa Rao. 2006-02-27. Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series. https://arxiv.org/abs/math/0602613

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