Three term relations for basic hypergeometric series

TL;DR

This paper derives explicit three-term linear relations for basic hypergeometric series {}_{2}phi_{1} with parameters differing by integer powers of q, providing formulas for the coefficients involved.

Contribution

It provides explicit formulas for the coefficients in three-term relations for {}_{2}phi_{1} series with specific parameter differences.

Findings

Explicit coefficient formulas for three-term relations.

Relations hold for parameters differing by integer powers of q.

Enhances understanding of hypergeometric series identities.

Abstract

Any three basic hypergeometric series {}_{2}phi_{1} whose respective parameters (a, b, c) differ by integer powers of the base q satisfy a linear relation with coefficients which are rational functions of a, b, c, q and the variable x. These relations are called three term relations for the basic hypergeometric series {}_{2}phi_{1}. This paper gives explicit expressions for the coefficients of these three term relations.