The Kontorovich-Lebedev transform as a map between $d$-orthogonal polynomials

Abstract

A slight modification of the Kontorovich-Lebedev transform is an automorphism on the vector space of polynomials. The action of this $K{L}_{\alpha }$-transform over certain polynomial sequences will be under discussion, and a special attention will be given the d-orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the $K{L}_{\alpha }$-transform of a 2-orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose $K{L}_{\alpha }$-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.

27 pages