Elliptic Hypergeometric Solutions to Elliptic Difference Equations

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.