Ladder operators and a second--order difference equation for general discrete Sobolev orthogonal polynomials

Abstract

We consider a general discrete Sobolev inner product involving the Hahn difference operator, so this includes the well-known difference operators D q and ∆ and, as a limit case, the derivative operator. The objective is twofold. On the one hand, we construct the ladder operators for the corresponding nonstandard orthogonal polynomials and we obtain the second-order difference equation satisfied by these polynomials. On the other hand, we generalise some related results appeared in the literature as we are working in a more general framework. Moreover, we will show that all the functions involved in these constructions can be computed explicitly.

On second order q-difference equations satisfied by Al-Salam-Carlitz I-Sobolev type polynomials of higher order

“…Having said all that, and to the best of our knowledge, an arbitrary number of q-derivatives acting at the same time on the two boundaries of a bounded orthogonality interval, has never been previously considered in the literature, and the present work is intended to be a first step in this direction. This reveals some small differences of the corresponding polynomial sequences, for example related with the parity of the polynomials, with respect to what happens considering only one mass point (as in [13]), and that we have right now under study. To be more precise, this paper deals with the sequence of monic q-polynomials {U (a) n (x; q, j)} n≥0 , orthogonal with respect to the Sobolev-type inner product f, g λ,µ = 1 a f (x; q)g(x; q)(qx, a −1 qx; q) ∞ d q x…”

“…In the present study, we consider an arbitrary number j ∈ N, j ≥ 1 of q-derivatives in the discrete part of the inner product. For an interesting related work to this paper, see for example the preprint [13], which appeared just a few days ago while we were giving the finishing touches to the present manuscript. There, the authors generalize the action of an arbitrary number of q-derivatives for general orthogonality measures, using the same techniques as for example in [14], and also in the present paper.…”

The annihilation operator for certain family of q-Hermite Sobolev-type orthogonal polynomials

“…Furthermore, q-derivatives can also appear involved in the discrete part of the measure, thus modifying the continuous part and giving rise to Sobolev-type perturbations of the q-Hermite I polynomials. For a recent and comprehensive study on general discrete Sobolev orthogonal polynomials including difference operators see [11].…”