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

A slight modification of the Kontorovich–Lebedev transform is an auto‐morphism on the vector space of polynomials. The action of this KL α ‐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 KL α ‐transform of a 2‐orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose KL α ‐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.