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

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.