An Expansion Formula of Basic Hypergeometric Series via the (1-xy,y-x)--Inversion and Its Applications

Jin Wang, Xinrong Ma

Published 2013-01-16, updated 2017-09-06Version 4

With the use of the (f,g)-inversion formula, we establish an expansion of basic hypergeometric ${}_{r}\phi_{s}$ series in argument $x\,t$ as a linear combination of ${}_{r+3}\phi_{s+2}$ series in $t$ and its various specifications. These expansions can be regarded as common generalizations of Carlitz's, Liu's, and Chu's expansion in the setting of $q$--series. As direct applications, some new transformation formulas of $q$--series including new approach to Askey--Wilson polynomials, the Rogers--Fine identity, Ramanujan's reciprocal theorem and Ramanujan's ${}_1\psi_1$ summation formula, as well as the well--poised Bailey lemma, are also obtained.