Orthogonal polynomials satisfying higher-order difference equations
The author constructs measures with orthogonal polynomials satisfying higher-order (\(>2\)) difference equations with coefficients independent of the degree of the polynomials. These measures are constructed by multiplying the discrete classical weights for the Charlier, Meixner, Krawtchouk and Hahn polynomials by certain well-chosen variants of annihilator polynomials of a finite set of numbers. This leads to some explicit examples of fourth-order difference equations satisfied by generalizations of the discrete classical orthogonal polynomials named after Charlier, Meixner, Krawtchouk and Hahn. Moreover, in the Charlier and Meixner cases, an explicit example is given of a sixth-order difference equation. The method leads to a number of conjectures. Only for the first conjecture (concerning the Meixner and the Hahn weight), a proof is presented for a special case of a generalization of the Meixner weight. More general proofs of this and other conjectures are announced to be given in (at least) two forthcoming papers.
scientific article; zbMATH DE number 6132831
scientific article