Bm$(1-xy,y-x)$-expansion formula and its applications

Abstract

In this paper, we first show a $(1-xy,y-x)$-expansion formula for an arbitrary analytic function $F(x)$ with respect to the basis <disp-formula><tex-math><![CDATA[$$\bigg\{\prod_{k=0}^{n-1}\frac{x-b_k}{1-x_kx}\,\bigg|\, n\geq 0\bigg\},$$]]></tex-math></disp-formula> which is mainly based on the $(f,g)$-inversion formula. Subsequently, by specializing $F(x)$, $x_n$ and $b_n$, we not only find the corresponding proofs of many classical results such as the Rogers-Fine identity, Andrews' four parametric reciprocal theorem, and Ramanujan's ${~}_1\psi_1$ summation formula, but also construct a lot of $q$-series transformations and summation formulas, including a generalization of Andrews' WP Bailey lemma.