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

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.