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 inline image-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 inline image-transform of a 2-orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose inline image-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.