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

TL;DR

This paper explores how a modified Kontorovich-Lebedev transform acts as an automorphism on polynomial spaces, specifically transforming certain polynomial sequences into d-orthogonal ones, and characterizes all such sequences.

Contribution

It introduces a modified KL-transform that maps polynomial sequences to d-orthogonal polynomials and characterizes all sequences with this property, revealing connections to semiclassical polynomials.

Findings

Continuous Dual Hahn polynomials are the KL-transform of Laguerre-type sequences.

All polynomial sequences mapped to d-orthogonal sequences are essentially semiclassical.

Hermite and Laguerre polynomials are classical solutions to the transformation problem.

Abstract

A slight modification of the Kontorovich-Lebedev transform is an automorphism on the vector space of polynomials. The action of this $KL_{\alpha}$-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 $KL_{\alpha}$-transform of a 2-orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose $KL_{\alpha}$-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.