Difference equations and Sobolev orthogonal polynomials, generated by Meixner polynomials

The representation of the Cauchy problem's solution for a difference equation with variable coefficients and given initial conditions at $x = 0$ by expanding this solution in a Fourier series on Sobolev polynomials orthogonal on the grid $(0,1,\ldots)$. The representation is based on contraction new polynomials orthogonal on Sobolev and generated by classical Meixner's polynomials. For new polynomials an explicit formula containing Meixner polynomials is obtained. This result allows us to investigate the asymptotic properties of new polynomials orthogonal on Sobolev on the grid $(0,1, \ldots)$ with a given weight. In addition, it allows to solve the problem of the calculation of the polynomials orthogonal on Sobolev, reducing it to use of well known recurrence relations for classical Meixner polynomials.