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

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 <tex-math><![CDATA[\∏ₖ₌₀ⁿ⁻¹x-bₖ/1-xₖx\,|\, n≥ 0\,]]></tex-math> which is mainly based on the $(f,g)$-inversion formula. Subsequently, by specializing $F(x)$, xₙ and bₙ, 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 ~₁ψ₁ summation formula, but also construct a lot of q-series transformations and summation formulas, including a generalization of Andrews' WP Bailey lemma.