Coxeter group actions on 4F3(1) hypergeometric series

TL;DR

This paper explores the symmetries and relations of a specific hypergeometric series, revealing invariance under certain group actions and deriving a large set of three-term relations classified by Hamming type.

Contribution

It establishes the invariance of a hypergeometric series under a symmetric group action and develops an algebra of three-term relations classified by Coxeter group symmetries.

Findings

Invariance of $K(a;b,c,d;e,f,g)$ under a group isomorphic to $S_6$.

Derivation of 4960 three-term relations classified by Hamming type.

Explicit examples of the five types of three-term relations provided.

Abstract

We investigate a certain linear combination $K(\vec{x})=K(a;b,c,d;e,f,g)$ of two Saalschutzian hypergeometric series of type ${_4}F_3(1)$. We first show that $K(a;b,c,d;e,f,g)$ is invariant under the action of a certain matrix group $G_K$, isomorphic to the symmetric group $S_6$, acting on the affine hyperplane $V=\{(a,b,c,d,e,f,g)\in\Bbb C^7\colon e+f+g-a-b-c-d=1\}$. We further develop an algebra of three-term relations for $K(a;b,c,d;e,f,g)$. We show that, for any three elements $\mu_1,\mu_2,\mu_3$ of a certain matrix group $M_K$, isomorphic to the Coxeter group $W(D_6)$ (of order 23040), and containing the above group $G_K$, there is a relation among $K(\mu_1\vec{x})$, $K(\mu_2\vec{x})$, and $K(\mu_3\vec{x})$, provided no two of the $\mu_j$'s are in the same right coset of $G_K$ in $M_K$. The coefficients in these three-term relations are seen to be rational combinations of gamma and sine functions in $a,b,c,d,e,f,g$. The set of $({|M_K|/|G_K|\atop 3})=({32\atop 3})=4960$ resulting three-term relations may further be partitioned into five subsets, according to the Hamming type of the triple $(\mu_1,\mu_2,\mu_3) $ in question. This Hamming type is defined in terms of Hamming distance between the $\mu_j$'s, which in turn is defined in terms of the expression of the $\mu_j$'s as words in the Coxeter group generators. Each three-term relation of a given Hamming type may be transformed into any other of the same type by a change of variable. An explicit example of each of the five types of three-term relations is provided.