The Kontorovich-Lebedev transform as a map between dd-orthogonal polynomials

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.Comment: 27 page

Central factorials under the Kontorovich-Lebedev transform of polynomials

We show that slight modifications of the Kontorovich-Lebedev transform lead to an automorphism of the vector space of polynomials. This circumstance along with the Mellin transformation property of the modified Bessel functions perform the passage of monomials to central factorial polynomials. A special attention is driven to the polynomial sequences whose KL-transform is the canonical sequence, which will be fully characterized. Finally, new identities between the central factorials and the Euler polynomials are found.Comment: also available at http://cmup.fc.up.pt/cmup/ since the 2nd August 201

Uncertainty principles for the kontorovich‐lebedev transform

By using classical uncertainty principles for the Fourier transform and composition properties of the Kontorovich‐Lebedev transform, analogs of the Hardy, Beurling, Cowling‐Price, Gelfand‐Shilov and Donoho‐Stark theorems are obtained. First Published Online: 14 Oct 201

Markov processes related to the stationary measure for the open KPZ equation

Motivated by recent results of Corwin and Knizel on stationary measures for the open KPZ equation on the spatial interval [0, 1], we study a pair of Markov processes with Laplace transforms that have dual representations, with the arguments of the Laplace transforms and the time parameters of the processes swapped. Combined with the results of Corwin and Knizel, our formula identifies the law of the stationary solutions for the open KPZ in terms of a Markov process which is a Doob's h transform of the Brownian motion killed at an exponential rate.Comment: Expanded versio