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

By R. Jagannathan, K. Rao, Shanmugha Arts. 2006

142 citations recorded by the source corpus.

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 −q n )/(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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cf939c184acdfa3bd200a88e38d5708bf4654134

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Citations in the corpus

On a Kantorovich Variant of (p,q) -Szász-Mirakjan Operators (2015, background): “Details on (p, q)-calculus can be found in [5, 8, 17].”
(p,q)‐Generalization of Szász–Mirakyan operators (2015, background): “For more details on (p, q)-calculus, we refer the readers to [5, 8, 7, 14, 15] and references therein.”
$$\left( p,q\right) $$p,q-Hermite–Hadamard inequalities and $$\left( p,q\right) $$p,q-estimates for midpoint type inequalities via convex and quasi-convex functions (2018, background): “1), then 0Dp,q f (t) = Dp,q f (t), where Dp,q f (t) is familiar (p, q)-derivative of f at t ∈ [a, b] defined by the expression (see [5,13]) Dp,q f (t) = f (pt) − f (qt) (p − q) t , t = 0.”