### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 026, 29 pages      arXiv:1310.7273      https://doi.org/10.3842/SIGMA.2014.026
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

### Symmetry Groups of $A_n$ Hypergeometric Series

Yasushi Kajihara
Department of Mathematics, Kobe University, Rokko-dai, Kobe 657-8501, Japan

Received September 30, 2013, in final form March 04, 2014; Published online March 18, 2014

Abstract
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.

Key words: multivariate hypergeometric series; elliptic hypergeometric series; Coxeter groups.

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