The little q-jacob1 functions of complex order

Abstract

We use Ismail's argument and an elementary combinatorial identity to prove the q-binomial theorem, the symmetry of the Rogers-Fine function, Ramanujan's 1Ψ1 sum, and Heine's q-Gauss sum and give many other proofs of these results. We prove a special case of Heine's 2ζ1 transformation and write Ramanujan's 1Ψ1 sum as the nonterminating q-Chu-Vandermonde sum. We show that the q-Saalschiitz and q-Chu-Vandermonde sums are equivalent to the evaluations of certain moments and to the orthogonality of the little q-Jacobi polynomials; hence the q-Chu-Vandermonde sum implies the q-Saalschütz sum. We extend the little q-Jacobi polynomials naturally to the little q-Jacobi functions of complex order. We show that the nonterminating q-Saalschütz and q-Chu-Vandermonde sums are equivalent to the evaluations of certain moments and, using the Liouville-Ismail argument, to two orthogonality relations. We show that the nonterminating q-Chu-Vandermonde sum implies the nonterminating q-Saalschütz sum. © 2005 Springer Science+Business Media, Inc.