# Coxeter group actions and limits of hypergeometric series

## TL;DR

This paper explores how Coxeter group actions relate to limits of hypergeometric series, revealing connections between different series types through group symmetries and limiting processes.

## Contribution

It demonstrates a novel approach linking hypergeometric series transformations with Coxeter group actions via limiting procedures, unifying various series types under group symmetries.

## Key findings

- Limiting processes connect $M$, $J$, and $L$ hypergeometric functions.
- Group actions on $M$ functions induce known symmetries on $J$ and $L$ functions.
- Three-term relations among functions are preserved under limits.

## Abstract

In this paper, we use combinatorial group theory and a limiting process to connect various types of hypergeometric series, and of relations among such series. We begin with a set $S$ of 56 distinct translates of a certain function $M$, which takes the form of a Barnes integral, and is expressible as a sum of two very-well-poised $_9F_8$ hypergeometric series of unit argument. We consider a known, transitive action of the Coxeter group $W(E_7)$ on this set. We show that, by removing from $W(E_7)$ a particular generator, we obtain a subgroup that is isomorphic to $W(D_6)$, and that acts intransitively on $S$, partitioning it into three orbits, of sizes 32, 12, and 12 respectively. Taking certain limits of the $M$ functions in the first orbit yields a set of 32 $J$ functions, each of which is a sum of two Saalsch\"utzian $_4F_3$ hypergeometric series of unit argument. The original action of $W(D_6)$ on the $M$ functions in this orbit is then seen to correspond to a known action of this group on this set of $J$ functions. In a similar way, the image of each of the size-12 orbits, under a similar limiting process, is a set of 12 $L$ functions that have been investigated in earlier works. In fact, these two image sets are the same. The limiting process is seen to preserve distance, except on pairs consisting of one $M$ function from each size-12 orbit. Finally, each known three-term relation among the $J$ and $L$ functions is seen to be obtainable as a limit of a known three-term relation among the $M$ functions.