Semyon Yakubovich

Multiple orthogonal polynomials associated with Macdonald functions

We consider multiple orthogonal polynomials corresponding to two Macdonald functions (modified Bassel functions of the second kind), with emphasis on the polynomials on the diagonal of the Hermite-Padé table. We give some properties of these polynomials: differential properties, a Rodrigues type formula and explicit formulas for the third order linear recurrence relation.

On the least values of L p -norms for the Kontorovich-Lebedev transform and its convolution

We establish analogs of the Hausdorff-Young and Riesz-Kolmogorov inequalities and the norm estimates for the Kontorovich-Lebedev transformation and the corresponding convolution. These classical inequalities are related to the norms of the Fourier convolution and the Hilbert transform in Lp spaces, 1 ≤ p ≤ ∞. Boundedness properties of the Kontorovich-Lebedev transform and its convolution operator are investigated. In certain cases the least values of the norm constants are evaluated. Finally, it is conjectured that the norm of the Kontorovich-Lebedev operator Ki τ : Lp (R+;xdx) → Lp(R+;x sinh πx dx), 2 ≤ p ≤ ∞ Kiτ[f] = ∫0∞ Kiτ(x)f(x)dx, τ ∈ R+ is equal to π/21-1/p. It confirms, for instance, by the known Plancherel-type theorem for this transform when p = 2.

Uncertainty principles for the kontorovich‐lebedev transform

Abstract 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.

Fundamental solutions of the fractional two-parameter telegraph equation

This paper is intended to investigate a fractional telegraph equation of the form with positive real parameters a, b and c. Here , and are operators of the Riemann–Liouville fractional derivative, where 0<α≤1 and 0<β≤1. A symbolic operational form of the solutions in terms of the Mittag–Leffler functions is exhibited. Using the Banach fixed point theorem, the existence and uniqueness of solutions are studied for this kind of fractional differential equations.

A class of polynomials and discrete transformations associated with the Kontorovich–Lebedev operators

We consider a class of polynomials related to the kernel K iτ(x) of the Kontorovich–Lebedev transformation. Algebraic and differential properties are investigated and integral representations are derived. We draw a parallel and establish a relationship with the Bernoullis and Eulers numbers and polynomials. Finally, as an application we invert a discrete transformation with the introduced polynomials as the kernel, basing it on a decomposition of Taylors series in terms of the Kontorovich–Lebedev operator.

Convolutions related to the Fourier and Kontorovich–Lebedev transforms revisited

We continue to investigate boundedness properties in a two-parametric family of Lebesgue spaces for convolutions related to the Fourier and Kontorovich–Lebedev transforms. Norm estimations in the weighted L p -spaces are obtained and applications to the corresponding class of convolution integral equations are demonstrated. Necessary and sufficient conditions are found for the solvability of these equations in the weighted L 2-spaces.

On a Riemann-Liouville fractional analog of the Laplace operator with positive energy

The purpose of this paper is to define a particular fractional analog of the Laplace operator in a rectangular domain in the plane by exploiting the Riemann–Liouville fractional derivatives. Such a definition allows the introduction of fractional boundary value problems which correspond to the classical Dirichlet, Neumann and mixed boundary value problems for the Laplace operator. By exploiting a suitable Integration by Parts Formula and the positiveness of the corresponding energy integral, we verify some uniqueness results for the solutions of the boundary value problems and show the existence of particular solutions.

Properties of index transforms in modeling of nanostructures and plasmonic systems

In material structures with nanometer scale curvature or dimensions, electrons may be excited to oscillate in confined spaces. The consequence of such geometric confinement is of great importance in nano-optics and plasmonics. Furthermore, the geometric complexity of the probe-substrate/sample assemblies of many scanning probe microscopy experiments often poses a challenging modeling problem due to the high curvature of the probe apex or sample surface protrusions and indentations. Index transforms such as Mehler–Fock and Kontorovich–Lebedev, where integration occurs over the index of the function rather than over the argument, prove useful in solving the resulting differential equations when modeling optical or electronic response of such problems. By considering the scalar potential distribution of a charged probe in the presence of a dielectric substrate, we discuss certain implications and criteria of the index transform and prove the existence and the inversion theorems for the Mehler–Fock transform ...

Eigenfunctions and Fundamental Solutions of the Fractional Two-Parameter Laplacian

We deal with the following fractional generalization of the Laplace equation for rectangular domains , which is associated with the Riemann-Liouville fractional derivatives , , where , . Reducing the left-hand side of this equation to the sum of fractional integrals by and , we then use the operational technique for the conventional right-sided Laplace transformation and its extension to generalized functions to describe a complete family of eigenfunctions and fundamental solutions of the operator in classes of functions represented by the left-sided fractional integral of a summable function or just admitting a summable fractional derivative. A symbolic operational form of the solutions in terms of the Mittag-Leffler functions is exhibited. The case of the separation of variables is also considered. An analog of the fractional logarithmic solution is presented. Classical particular cases of solutions are demonstrated.

Closed-form evaluation of two-dimensional static lattice sums

Closed-form formulae for the conditionally convergent two-dimensional (2D) static lattice sums S2 (for conductivity) and T2 (for elasticity) are deduced in terms of the complete elliptic integrals of the first and second kind. The obtained formulae yield asymptotic analytical formulae for the effective tensors of 2D composites with circular inclusions up to the third order in concentration. Exact relations between S2 and T2 for different lattices are established. In particular, the value S2=π for the square and hexagonal arrays is discussed and T2=π/2 for the hexagonal is deduced.