Symmetries of coefficients of three-term relations for basic hypergeometric series

TL;DR

This paper investigates the symmetries of the coefficients in three-term relations for basic hypergeometric series, revealing ninety-six symmetries and providing explicit formulas for them, advancing understanding of their algebraic structure.

Contribution

The paper proves that the coefficients of three-term relations for ${}_{2}\phi_{1}$ have ninety-six symmetries and explicitly describes these symmetries.

Findings

Identified ninety-six symmetries of the coefficients.

Provided explicit formulas for the symmetries.

Enhanced understanding of the algebraic structure of hypergeometric relations.

Abstract

Any three basic hypergeometric series ${}_{2}\phi_{1}$ whose respective parameters $a, b, c$ and a variable $x$ are shifted by integer powers of $q$ are linearly related with coefficients that are rational functions of $a, b, c, q$, and $x$. This relation is called a three-term relation for ${}_{2}\phi_{1}$. In this paper, we prove that the coefficients of the three-term relation for ${}_{2}\phi_{1}$ considered in the author's earlier paper (2022) have ninety-six symmetries, and present explicit formulas describing these symmetries.