Contiguous relations and summation and transformation formulas for basic hypergeometric series

TL;DR

This paper uses contiguous relations to provide simplified proofs of various summation and transformation formulas for basic hypergeometric series, also deriving finite forms of classical identities.

Contribution

It introduces a unified approach using contiguous relations to prove multiple hypergeometric identities and derives finite forms of classical identities.

Findings

Simplified proofs of Bailey's, Carlitz's, Sears', Chen's, Gasper's, and Chu's identities.

Finite forms of Sylvester's, Jacobi's, and Kang's identities.

New transformations and summations for basic hypergeometric series.

Abstract

By using contiguous relations for basic hypergeometric series, we give simple proofs of Bailey's $_4\phi_3$ summation, Carlitz's $_5\phi_4$ summation, Sears' $_3\phi_2$ to $_5\phi_4$ transformation, Sears' ${}_4\phi_3$ transformations, Chen's bibasic summation, Gasper's split poised $_{10}\phi_9$ transformation, Chu's bibasic symmetric transformation. Along the same line, finite forms of Sylvester's identity, Jacobi's triple product identity, and Kang's identity are also obtained.