P,Q-differentiation, P,Q-integration, and P,Q-hypergeometric functions related to quantum groups

Abstract

Investigation of representations of two-parametric quantum groups leads to two-parametric differentiation and integration which are generalizations of the well known q-differentiation and q-integration. In this paper these differentiation and integration are studied. Their connection with q-differentiation and q-integration is derived. The p,q-hypergeometric functions are introduced. Their relation to the basic hypergeometric functions is studied. It is emphasized that investigation of some operators of representations of quantum algebras leads to orthogonal polynomials determining their spectral measures and their spectra. Operators of representations of the quantum algebra U_pq :(su) and of the p,q-oscillator algebra are studied. Some of these operators lead to unknown orthogonal polynomials. Other ones give particular cases of q-Askey-Wilson polynomials and g-Hermite polynomials with the base r = (pq) ^1/2.