q-Series and Orthogonal Polynomials Associated with Barnes’ First Lemma

We exploit symmetry (recurrence relation) techniques for the derivation of properties associated with families of basic hypergeometric functions. Similar methods have been used by Nikiforov, Suslov, and Uvarov. Here we apply these ideas to find new proofs of Barnes’ First Lemma and some of its q-analogues. We show that these integrals correspond to the weight functions determining the orthogonality relations for Hahn, q-Hahn, and big q-Jacobi polynomials. As another example of our method we introduce a biorthogonal system of rational functions whose weight function corresponds to the q-analogue of Kummer’s Theorem.