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 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.

On the orthogonality and convolution orthogonality via the Kontorovich-Lebedev transform

“…With a slight modification of the Kontorovich-Lebedev transform (1.1) it becomes an automorphism on the vector space of polynomials P (see [7], [8]). Moreover, it involves orthogonal and multiple orthogonal polynomials or d-orthogonal ones.…”

‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers

“…Remark 3. It is worth to notice that the change of basis between {(a + ıx) n (a − ıx) n } n 0 and {(x 2 + (a − 1) 2 ) n } n 0 is performed by the aforementioned pair of Jacobi-Stirling numbers { (−1) n+k js k+1 n+1 (2α), JS k+1 n+1 (2α) } 0 k n , see [25,Remark 3.9]. Indeed, the latter pair of numbers is associated to the linear change of basis in the vector space of polynomials in the variable x 2 performed by the Kontorovich-Lebedev integral transform after a slight modification of the kernel, as described in [25].…”