Symmetry Groups of $A_n$ Hypergeometric Series

Structures of symmetries of transformations for Holman–Biedenharn–Louck $A_n$ hypergeometric series: $A_n$ terminating balanced ${}_4 F_3$ series and $A_n$ elliptic ${}_{10} E_9$ series are discussed. Namely the description of the invariance groups and the classification all of possible transformations for each types of $A_n$ hypergeometric series are given. Among them, a “periodic” affine Coxeter group which seems to be new in the literature arises as an invariance group for a class of $A_n$ ${}_4 F_3$ series.