Transformations and summations for bilateral basic hypergeometric series

TL;DR

This work develops a comprehensive framework of transformation and summation formulas for bilateral basic hypergeometric series, driven by two foundational ${}_8\Psi_8$ transformations. By employing limiting processes, the authors derive extensive families of very-well-poised bilateral identities for lower-order series (${\ }_{7}\Psi_{7}^{1}$ down to ${}_2\Psi_2^{6}$) and establish a broad network of implied transformations for many non-very-well-poised bilateral series. They also connect these results to high-order tuple product identities, expressing them as sums of bilateral series, thereby enriching the connections between product identities and bilateral q-hypergeometric theory. The paper corrects and consolidates key transformations (notably Zhang–Zhang and Wei–Yu) and provides a valuable catalog of identities that can serve as a resource for q-series researchers and related combinatorial applications.

Abstract

We derive transformation and summation formulas for bilateral basic hypergeometric series. As a starting point, we use two transformations of bilateral basic very-well-poised ${}_8Ψ_8$. The first transformation is given as a sum of two nonterminating ${}_8W_7$'s and the second is given in terms of a sum of a ${}_4ψ_4$ and two balanced ${}_4φ_3$'s. From these transformations we derive limiting transformations with vanishing denominator elements which shed light on the transformation properties of these bilateral basic hypergeometric series. We also study tuple product identities, namely triple, quintuple, sextuple, septuple, octuple, nonuple and undecuple, which are given in terms of sums of bilateral basic hypergeometric series.

Theorems & Definitions (59)

• Definition 1.1
• Lemma 1.2
• Remark 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Theorem 3.1
• Theorem 3.2
• Remark 3.3
• Theorem 3.4