Quantum Algebras and q-Special Functions

Abstract A quantum-algebraic framework for many q-special functions is provided. The two-dimensional Euclidean quantum algebra, slq(2) and the q-oscillator algebra are considered. Realizations of these algebras in terms of operators acting on vector spaces of functions in one complex variable are given. Basic hypergeometric functions are shown to arise, in analogy with Lie theory, as matrix elements of certain operators. New generating functions for these q-special functions are obtained.