On discrete q-ultraspherical polynomials and their duals

Abstract

A special case of the big q-Jacobi polynomials P n ( x ; a , b , c ; q ) , which corresponds to a = b = − c , is shown to satisfy a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0 < a < q −1 ). Since P n ( x ; q α , q α , − q α ; q ) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q → 1 , this family represents another q-extension of these classical polynomials, different from the continuous q-ultraspherical polynomials of Rogers. For a dual family with respect to the polynomials P n ( x ; a , a , − a ; q ) (i.e., for dual discrete q-ultraspherical polynomials) we also find new orthogonality relations with extremal measures.