ON THE RELATION BETWEEN GLOBAL PROPERTIES OF LINEAR DIFFERENCE AND DIFFERENTIAL-EQUATIONS WITH POLYNOMIAL COEFFICIENTS .1.

AbstractThis paper is concerned with applications of the Mellin transformation in the study of homogeneous linear differential and difference equations with polynomial coefficients. We begin by considering a differential equation (D) with regular singularities at O and ∞ and arbitrary singularities in the rest of the complex plane, and the difference equation (Δ′) obtained from (D) by a variant of the formal Mellin transformation. We define fundamental systems of solutions of (Δ′), analytic in either a right or a left half plane. by the use of Mellin transforms of microsolutions of (D). The relations between these fundamental systems are expressed in terms of central connection matrices of (D). Second, we study the differential equation (D1) obtained from (D) by means of a formal Laplace transformation and the difference equation (Δ1) obtained from (D1) by a formal Mellin transformation. We use Mellin transforms of "ordinary" solutions of (D1) with moderate growth at ∞ to construct fundamental systems of solutions of (Δ1). The relation between these fundamental systems involves certain Stokes multipliers and a formal monodromy matrix of (D1)

Summation of formal solutions of a class of linear difference equations

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