Some expansion formulas for q-series and their applications

Abstract

In this paper, we establish some general expansion formulas for q-series. Three of Liu's identities motivate us to search and find such type of formulas. These expansion formulas include as special cases or limiting cases many q-identities including the q-Gauss summation formula, the q-Pfaff-Saalschütz summation formula, three of Jackson's transformation formulas and Sears' terminating ${}_4\varphi _3$ transformation formula. As applications, we provide a new proof of the orthogonality relation for continuous dual q-Hahn polynomials, establish some generating functions for special values of the Dirichlet L-functions and the Hurwitz zeta functions, give extensions of three of Liu's identities, establish an extension of Dilcher's identity, and deduce various double Rogers-Ramanujan type identities.