Telescoping partial fractions decompositions, the little q-Jacobi functions of complex order, and the nonterminating q-Saalschütz sum

Abstract

We use telescoping partial fractions decompositions to give new proofs of the orthogonality property and the normalization relation for the little q-Jacobi polynomials, and the q-Saalschutz sum. In [20], we followed the development [19] of Schur functions for partitions with complex parts, and we showed that there exist natural little q-Jacobi functions of complex order which satisfy extensions of the orthogonality property and normalization relation of the little q-Jacobi polynomials, and that these two results follow from and together imply the nonterminating form of the q-Saalschutz sum. Writing the q-Pochhammer symbol of complex order as a ratio of infinite products in the usual way, we obtain new telescoping partial fractions decomposition proofs of our results [20] for the little q-Jacobi functions of complex order. We give several new proofs of the q-Saalschutz sum and its nonterminating form.