Elliptic Hypergeometric Solutions to Elliptic Difference Equations

TL;DR

This paper explores solutions to first-order linear difference equations on elliptic lattices, revealing simple interpolatory expansions for solutions based on elliptic functions.

Contribution

It introduces a method to define and analyze first-order linear difference equations on elliptic lattices using elliptic hypergeometric functions.

Findings

Solutions have simple interpolatory expansions.

Focus on difference equations on elliptic lattices.

Provides a framework for elliptic hypergeometric solutions.

Abstract

It is shown how to define difference equations on particular lattices $\{x_n\}$, $n\in\mathbb{Z}$, made of values of an elliptic function at a sequence of arguments in arithmetic progression (elliptic lattice). Solutions to special difference equations have remarkable simple interpolatory expansions. Only linear difference equations of first order are considered here.