Difference operators and difference equations on lattices, or grids, up to the elliptic hypergeometric case

Abstract

It is shown how to define difference operators and equations on particular lattices $\{x_n\}$, $2n\in\mathbb{Z}$, such that the divided difference operator $(\mathcal{D}f)(x_{n+1/2})= (f(x_{n+1})-f(x_n))/(x_{n+1}-x_n)$ has the property that $\mathcal{D}f$ is a rational function of degree $2d$ when $f$ is a rational function of degree $d$. It is then shown that the $x_n$s are in the most general case values of an elliptic function at a sequence of arguments in arithmetic progression (\emph{elliptic lattice}). Many special and limit cases, down to the most elementary ones, are considered too. First and second order difference operators and equations are constructed, up to the simplest elliptic hypergeometric ones. One also shows orthogonality and biorthogonality properties of rational solutions to some of these difference equations.