Summation of formal solutions of a class of linear difference equations

Abstract

We consider difference equations y(s+1) = A(s)y(s), where A(s) is an n x n-matrix meromorphic in a neighborhood of infinity with det A(s) not equal 0. In general, the formal fundamental solutions of this equation involve gamma-functions which give rise to the critical variable s log s and a level 1(+). We show that, under a mild condition, formal fundamental matrices of the equation can be summed uniquely to analytic fundamental matrices represented asymptotically by the formal fundamental solution in appropriate domains. The method of proof is analogous to a method used to prove multi-summability of formal solutions of ODE's. Starting from analytic lifts of the formal fundamental matrix in half planes, we construct a sequence of increasingly precise quasi-functions, each of which is determined uniquely by its predecessor.