Polynomials Orthogonal with Respect to Sobolev Type Inner Product Generated by Charlier Polynomials

The problem of constructing of the Sobolev orthogonal polynomials s(r,n)(alpha)(x) generated by Charlier polynomials s(n)(alpha)(x) is considered. It is shown that the system of polynomials s(r,n)(alpha)(x) generated by Charlier polynomials is complete in the space W-l rho(r), consisted of the discrete functions, given on the grid Omega = {0, 1, ...}. W-l rho(r) is a Hilbert space with the inner product < f, g >. An explicit formula in the form of s(r,k+r)(alpha)(x) = Sigma(k)(l=0) b(l)(r)x([l+r]),where x([m]) = x(x - 1) (x - m + 1), is found. The connection between the polynomials s(r,n)(alpha)(x) and the classical Charlier polynomials s(n)(alpha)(x) in the form of s(r,k+r)(alpha)(x) = U-k(r )[s(k+r)(alpha)(x) - Sigma(r-1)(nu=0)V(k,nu)(r )x([)(nu])], where for the numbers U-k(r), V-k,nu(r) we found the explicit expressions, is established.