With the use of the $(f,g)$-inversion formula under specializations that$f=1-xy,g=y-x$, we establish an expansion of (modified) basic hypergeometric${}_{r}\phi_{s}$ series in variable $x~t$ as a linear combination of${}_{r+2}\phi_{s+1}$ series in $t$ and its various specifications. Theseexpansions can be regarded as common generalizations of Carlitz's, Liu's, andChu's expansion in the setting of $q$-series. As direct applications, some newtransformation formulas of $q$-series including new approach to theAskey-Wilson polynomials, the Rogers-Fine identity, Andrews' four-parametricreciprocity theorem and Ramanujan's ${}_1\psi_1$ summation formula, as well asa transformation for certain well-poised Bailey pairs, are presented.
展开▼
