Representations of two parameter quantum algebras and \(p,q\)-special functions (Q2642113)

The authors show the connection between \(p,q\)-analogue of special functions and representations of certain two parameter quantum algebras. Firstly, the authors collect some elements of \(p,q\)-special function theory and derive some new results. They introduce a generalized definition of \(p,q\)-hypergeometric series for an arbitrary number of numerator and denominator parameters. After these preliminaries, they discuss the quantum Euclidean algebra \(\xi_{p,q}(2)\), the \(p,q\)-oscillator algebra and the quantum algebra \(\text{sl}_{p,q}(2)\). They realize the generators of these algebras in terms of \(p,q\)-derivative operators in one complex variable and calculate the matrix elements corresponding to their representations in terms of the generalized \(p,q\)-hypergeometric series.