Representation of the solution of the Cauchy problem for a difference equation by a Fourier series in Meixner - Sobolev polynomials

We obtain a representation of the solution to the Cauchy problem for the $r$-th order difference equation with constant coefficients and given initial conditions at the point $x=0$. This representation is based on the expansion of the solution in the Fourier series by polynomials that are orthogonal in the sense of Sobolev on the grid $\{0, 1, \ldots\}$ and generated by the classical Meixner polynomials. In addition, an algorithm for numerical finding of the unknown coefficients in this expansion has been developed.