Two new transformation formulas for ${}_{8}\psi_{8}$ and ${}_{8}W_{7}$ series associated with Weierstrass' theta identity

TL;DR

This paper introduces two novel transformation formulas for specific basic hypergeometric series, leveraging Slater's bilateral series transformation, and applies them to derive a general Weierstrass theta identity and new proofs of classical summation formulas.

Contribution

It presents new transformation formulas for ${}_{8} heta_{8}$ and ${}_8 ext{W}_7$ series, expanding the toolkit for hypergeometric series analysis and providing alternative proofs for known identities.

Findings

Derived two new transformation formulas for ${}_{8} heta_{8}$ and ${}_8 ext{W}_7$ series.

Established a general form of Weierstrass' theta identity.

Provided new proofs for Bailey's ${}_6 heta_6$ and Jackson's ${}_8 ext{W}_7$ summation formulas.

Abstract

In this paper, we establish two new transformation formulas for ${}_{8}\psi_{8}$ and ${}_8\phi_7$ series by means of Slater's general transformation for bilateral series. As applications, some specific transformation formulas are presented among which include a general form of Weierstrass' theta identity and new proofs of Bailey's VWP ${}_6\psi_6$ and Jackson's ${}_8\phi_7$ summation formula.