Abstract We establish the boundedness properties in Lp for a class of integral transformations with respect to an index of hypergeometric functions. In particular, by using the RieszThorin interpolation theorem we get the corresponding results in Lp(R+), 1 ≤ p ≤ 2 for the Kontorovich-Lebedev, Mehler-Fock and Olevskii index transforms. An inversion theorem is proved for general index transformat...

Strong asymptotic results for relativistic Hermite polynomials H n (z) are established as n, N →∞, thereby supplementing recent results on weak asymptotics for these polynomials. Depending on growth properties of the ratio N/n for the rescaled polynomials H n (cnz) (cn being suitable positive numbers, n, N →∞), formulae of Plancherel-Rotach type are derived on the oscillatory interval, in the c...

Small perturbations of the Jacobi matrix with weights √ n and zero diagonal are considered. A formula relating the asymptotics of polynomials of the first kind to the spectral density is obtained, which is analogue of the classical Weyl-Titchmarsh formula for the Schrödinger operator on the half-line with summable potential. Additionally a base of generalized eigenvectors for ”free” Hermite ope...

In the first part of the paper kernels are constructed which meromorphically extend the Macdonald-Koornwinder polynomials in their degrees. In the second part of the paper the kernels associated with rank one root systems are used to define nonsymmetric variants of the spherical Fourier transform on the quantum SU(1, 1) group. Related Plancherel and inversion formulas are derived using double a...

We use discrete analogs of Riemann–Hilbert problem’s methods to derive the discrete Bessel kernel which describes the poissonized Plancherel measures for symmetric groups. To do this we define a discrete analog of 2 by 2 Riemann–Hilbert problems of special type. We also give an example, explicitly solvable in terms of classical special functions, when a discrete Riemann–Hilbert problem converge...

Abstract. The focus of this paper is weighted uncertainty principle inequalities in harmonic analysis. We start by reviewing the classical uncertainty principle inequality, and then proceed to extensions and refinements by modifying two major results necessary to prove the classical case. These are integration by parts and the Plancherel theorem. The modifications are made by means of generaliz...

We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble is related to a two-dimensional directed growth model, and the Charlier ensemble is related to the...