(1-xy,y-x)-展开公式与应用

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

本文首先利用(f, g)-反演公式建立了关于任意解析函数F(x)在给定基{^(n-1)∏(k=0)(x-b_(k))/(1-xkx)|n≥0}下所谓的(1-xy, y-x)-展开公式.随后,通过考虑具体的F(x)以及参数xn和bn,不但证明了很多经典结论,如Rogers-Fine恒等式、Andrews四参数互反定理、Ramanujan1ψ1求和公式,而且建立了大量的q-级数变换与求和公式,并且得到Andrews的WP Bailey引理的一种推广.

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 {^(n-1)∏(k=0)(x-b_(k))/(1-xkx)|n≥0},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.