The Kontorovich–Lebedev Transform as a Map between d‐Orthogonal Polynomials

Abstract

A slight modification of the Kontorovich–Lebedev transform is an auto‐morphism on the vector space of polynomials. The action of this ‐transform over certain polynomial sequences will be under discussion, and a special attention will be given to the d‐orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the ‐transform of a 2‐orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose ‐transform is a d‐orthogonal sequence will be characterized: they are essencially semiclassical polynomials fulfilling particular conditions and d is even. The Hermite and Laguerre polynomials are the classical solutions to this problem.