关于双边级数2ψ2.docVIP

关于双边级数 2ψ2 谷珊珊 南开大学组合数学中心，天津 300071 摘要：文章得到了一个 2ψ2 级数表示成两个 2?1 级数的恒等式，此等式可看作是 Slater 关于 2ψ2 级数公式的一个相伴形式。同时，通过双边扩展和参数扩充，Slater 的一个两项 2ψ2 级数 和式能够从 Andrews 的一个单边和式得到。 关键词：组合数学，基本超几何级数，双边级数，双边扩展，参数扩充。 中图分类号： O157 On the Bilateral Series 2ψ2 Nancy Shanshan Gu Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071 Abstract: A formula which reduces the evaluation of a 2ψ2 series to two 2?1 series is obtained. This identity may be considered as a companion of Slater’s formulas for 2ψ2 series. Meanwhile, a two-term 2ψ2 summation formula of Slater is obtained from a unilateral summation formula of Andrews by using bilateral extension and parameter augmentation. Key words: combinatorics, basic hypergeometric series, bilateral series, bilateral extension, parameter augmentation. 0 Introduction It is well known that many bilateral basic hypergeometric identities can be derived from unilateral identities. Using Cauchy’s method [1, 2, 3, 4] one may obtain bilateral basic hy- pergeometric identities from terminating unilateral identities. Starting with nonterminating unilateral basic hypergeometric series, Chen and Fu [5] developed a method to give semi-?nite forms by shifting the summation index by m. Then the bilateral summations are consequences of the semi-?nite forms by letting m tend to in?nity. We call this method bilateral extension. Jouhet [6] also obtained more semi-?nite forms of bilateral basic hypergeometric series by using this approach. In this paper we make use of the bilateral extension of a 3?2 series and an identity of Andrews [7] to study the bilateral series 2ψ2: 2ψ2 [  a, b c, d ] ; q, z .  (1) 基金项目： SRFDP (200800551042), NSFC 作者简介： Nancy Shanshan Gu（1980-），female，associate professor，major research direction：Basic hypergeometric series, Email: gu@nankai.edu.cn. -1- The above 2ψ2 series is closely related to the question of ?nding a q-extension of Dougall’s bilateral hypergeometric series summation formula [8]: ∞ k=?∞