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

TL;DR

This paper introduces a (p,q)-hypergeometric series framework that generalizes q-series using twin-basic numbers, providing new identities and simplifying derivations in quantum algebra and combinatorics.

Contribution

It develops a novel (p,q)-series extension of q-identities based on twin-basic numbers, expanding the toolkit for quantum algebra and combinatorial analysis.

Findings

Derived (p,q)-extensions of known q-identities

Showed (p,q)-series simplifies certain limiting processes

Connected twin-basic numbers to two-parameter quantum algebras

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.