Second-Order Difference Equation for Sobolev-Type Orthogonal Polynomials. Part II: Computational Tools

Abstract

We consider polynomials which are orthogonal with respect to a nonstandard inner product. In fact, we deal with Sobolev-type orthogonal polynomials in the broad sense of the expression. This means that the inner product under consideration involves the Hahn difference operator, thus including the difference operators $\mathscr{D}_{q}$ and $\Delta$ and, as a limit case, the derivative operator. In a previous work, we studied properties of these polynomials from a theoretical point of view. There, we obtained a second-order differential/difference equation satisfied by these polynomials. The aim of this paper is to present an algorithm and a symbolic computer program that provides us with the coefficients of the second-order differential/difference equation in this general context. To illustrate both, the algorithm and the program, we will show three examples related to different operators.