Quantum algebra approach to q Gegenbauer polynomials

Quantum algebras provide a natural algebraic setting for q special functions. The matrix elements of certain algebra generators in irreducible representations are in fact expressible in terms of q hypergeometric series. Taking the quantum algebra U((1,1)) as example, we shall show that its metaplectic representation provides a group-theoretic setting for certain basic orthogonal polynomials generalizing the usual Gegenbauer polynomials.