On minimal meromorphic solutions of difference equations

We consider difference equations with meromorphic coefficients in the complex plane. Assuming that the equations' coefficients are 1-periodic, we describe the minimal meromorphic solutions, i.e., the solutions that, under certain conditions on their poles, have the minimal possible growth at ±i∞. The notion of minimal meromorphic solution naturally arises in mathematical physics, for example, in the framework of the Sommerfeld-Malyuzhinets method and when studying difference equations of solid state physics.