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

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.

16 pages, To appear in the Proceedings of the International Conference on Number Theory and Mathematical Physics, 20-21 December 2005, Srinivasa Ramanujan Centre, Kumbakonam, India

References in corpus (1)

• (P,Q)-Special Functions

Cited by in corpus (26)

• (p,q)-Generalization of Szasz-Mirakyan Operators
• On the fundamental theorem of $(p,q)$-calculus and some $(p,q)$-Taylor formulas
• On the $(p,q)$-Gamma and the $(p,q)$-Beta functions
• Quantum deformed algebras : Coherent states and special functions
• R(p,q)-deformed conformal Virasoro algebra
• On (p, q) Baskakov-Durrmeyer-Stancu Operators
• Some Approximation Results by $(p,q)$-analogue of Bernstein-Stancu Operators (Revised)
• On (p,q)-analogue of Bernstein Operators (Revised)
• Approximation by (p,q)-Lorentz polynomials on a compact disk
• Some approximation results by Bernstein-Kantorovich operators based on (p,q)-integers
• Generalized Witt and Witt n-algebras, Virasoro algebras and constraints, and KdV equations from R(p,q)-deformed quantum algebras
• Approximation by Meyer-Konig and Zeller Operators using (p, q)-CALCULUS
• Generalized Heisenberg-Virasoro algebra and matrix models from quantum algebra
• Approximation by Kantorovich type (p,q)-Bernstein-Schurer Operators
• Generalized q-deformed oscillators, q-Hermite polynomials, generalized coherent states
• Statistical approximation by $(p,q)$-analogue of Bernstein-Stancu Operators
• On King Type Modification of $(p,q)$ Baskakov Operators which preserves $x^2$
• (p, q)-Szász-Mirakyan-Baskakov-Stancu type Operators
• New $q-$Hermite polynomials: characterization, operator algebra and associated coherent states
• On a Kantorovich variant of (p,q)-Szasz-Mirakjan operators
• R(p,q)- analogs of discrete distributions: general formalism and application
• R(p,q)-deformed combinatorics: full characterization and illustration
• On two $(p,q)$-analogues of the Laplace transform
• $(p,q)-$deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms
• Characterization of $(\mathcal{R},p,q)-$deformed Rogers-Szegö polynomials: associated quantum algebras, deformed Hermite polynomials and relevant properties
• Algorithms and Identities for B$\acute{e}$zier curves via Post Quantum Blossom