Infinite elliptic hypergeometric series: convergence and difference equations

We derive finite difference equations of infinite order for theta-hypergeometric series and investigate the space of their solutions. In general, such infinite series diverge, and we describe some constraints on the parameters when they do converge. In particular, we lift the Hardy-Littlewood criterion of the convergence of $q$-hypergeometric series for ${|q|=1}$, $q^n\neq 1$, to the elliptic level and prove the convergence of infinite very-well poised elliptic hypergeometric ${}_{r+1}V_r$-series for restricted values of $q$.
Bibliography: 13 titles.