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Semyon Yakubovich

Publications and source records attributed to Semyon Yakubovich.

At least 19 recordsLinked to original sources
arXiv2026

Generalized product formulas for Whittaker's functions and a novel class of index transforms

Generalized product formulas and index transforms, involving products of Whittaker's functions of different indices are established and investigated. The corresponding inversion formulas are found. Particular cases cover index transforms with products of the modified Bessel and Whittaker's functions. For our goals the Kontorovich-Lebedev and Olevskii transforms of a complex index with nonzero real part are involved.

math.CA↗
arXiv2026

Phoretic flow in a three-dimensional wedge geometry

Understanding how chemically induced surface transport generates fluid motion in confined geometries is essential for the rational design of microscale pumping devices and active microfluidic systems. Here we develop a theoretical framework for chemically driven phoretic flows in a three-dimensional wedge geometry in the diffusion-dominated regime. We formulate both the diffusion and hydrodynamic problems using a Fourier-Kontorovich-Lebedev spectral representation, exploiting the translational invariance and radial structure of the wedge. Green's functions for the concentration field are derived for reflecting and mixed reflecting-absorbing boundaries, reducing to finite image-like sums or closed-form expressions for commensurate wedge angles. The resulting slip velocity is then used to construct the three-dimensional Stokes flow through the Papkovich-Neuber representation, yielding explicit spectral solutions for the velocity field. These results establish a Green's-function framework for phoretic pumping in wedge-shaped confinement and provide analytical benchmarks for numerical simulations of chemically driven transport in confined microfluidic systems.

cond-mat.soft↗
arXiv2026

The fractional Kontorovich-Lebedev transform

We introduce the so-called fractional analog of the Kontorovich-Lebedev transform, modifying its kernel, which is the Macdonald function $K_ν(z)$. Properties of this kernel as well as mapping properties of the new index transform are investigated. The inversion formula in suitable spaces of functions is established.

math.CA↗
arXiv2026

On the Macdonald-type function and its relation with index transforms and orthogonal polynomials

We continue to investigate properties of the function $M_ν(z)$ which is associated with the Macdonald function $K_ν(z)$ in terms of the corresponding Fourier integral. In particular, recurrence relations for this function and its derivatives are obtained, involving properties of the associated Laguerre polynomials. Multiple orthogonal polynomials related to the scaled Macdonald-type weights $ \hatρ_ν(x)= 2 x^{ν/2} M_ν\left(2\sqrt x\right), x >0$ are investigated.

math.CA↗
arXiv2025

A new index transform with the square of Whittaker's function

An index transform, involving the square of Whittaker's function is introduced and investigated. The corresponding inversion formula is established. Particular cases cover index transforms of the Lebedev type with products of the modified Bessel functions.

math.CA↗
arXiv2024

Poisson summation formula and Index Transforms

Following the work of the second author, a class of summation formulas attached to index transforms is studied in this paper. Our primary results concern summation and integral formulas with respect to the second index of the Whittaker function $W_{μ,ν}(z)$.

math.CA↗
arXiv2023

Generalizations (in the spirit of Koshliakov) of some formulas from Ramanujan's Lost Notebook

In his lost notebook, Ramanujan recorded beautiful identities. These include earlier versions of Koshliakov's formula for the divisor function and the transformation formula for the logarithm of Dedekind's $η-$function. In this paper we establish some generalizations of these formulas of Ramanujan in a setting that only recently reemerged in the literature and which concerns a beautiful theory due to Koshliakov.

math.NT↗
arXiv2023

Certain Extensions of Results of Siegel, Wilton and Hardy

Recently, Dixit et al. established a very elegant generalization of Hardy's Theorem concerning the infinitude of zeros that the Riemann zeta function possesses at its critical line. By introducing a general transformation formula for the theta function involving the Bessel and Modified Bessel functions of the first kind, we extend their result to a class of Dirichlet series satisfying Hecke's functional equation. In the process, we also find new generalizations of classical identities in Analytic number theory.

math.NT↗
arXiv2022

Identities for combinatorial sums involving trigonometric functions

Let $$ A_{m,n}(a)=\sum_{j=0}^m (-4)^j {m+j\choose 2j}\sum_{k=0}^{n-1} \sin(a+2kπ/n) \cos^{2j}(a+2kπ/n) $$ and $$ B_{m,n}(a)=\sum_{j=0}^m (-4)^j {m+j+1\choose 2j+1}\sum_{k=0}^{n-1} \sin(a+2kπ/n) \cos^{2j+1}(a+2kπ/n), $$ where $m\geq 0$ and $n\geq 1$ are integers and $a$ is a real number. We present two proofs for the following results: (i) If $2m+1 \equiv 0 \, (\mbox{mod} \, n)$, then $$ A_{m,n}(a)=(-1)^m n \sin((2m+1)a). $$ (ii) If $2m+1 \not\equiv 0 \, (\mbox{mod} \, n)$, then $A_{m,n}(a)=0$. (iii) If $2(m+1) \equiv 0 \, (\mbox{mod} \, n)$, then $$ B_{m,n}(a)=(-1)^m \frac{n}{2} \sin(2(m+1)a). $$ (iv) If $2(m+1) \not\equiv 0 \, (\mbox{mod} \, n)$, then $B_{m,n}(a)=0$.

math.CA↗
arXiv2022

On the Epstein zeta function and the zeros of a class of Dirichlet series

By generalizing the classical Selberg-Chowla formula, we establish the analytic continuation and functional equation for a large class of Epstein zeta functions. This continuation is studied in order to provide new classes of theorems regarding the distribution of zeros of Dirichlet series in their critical lines and to produce a new method for the study of these problems. Due to the symmetries provided by the representation via the Selberg-Chowla formula, some generalizations of well-known formulas in analytic number theory are also deduced as examples.

math.NT↗
arXiv2021

A modification of the Prudnikov and Laguerre polynomials

A two-parameter sequence of orthogonal polynomials $\{P_n( x; λ, t)\}_{n\ge 0}$ with respect to the weight function $x^αe^{- λx} ρ_ν(x t),\ α> -1,\ λ, t \ge 0, \ ρ_ν(x)= 2 x^{ν/2} K_ν(2\sqrt x),\ x >0, ν\ge 0$, where $K_ν(z)$ is the modified Bessel function, is investigated. The case $λ=0$ corresponds to the Prudnikov polynomials and $t=0$ is related to the Laguerre polynomials. A special one-parameter case $\{P_n( x; 1-t, t)\}_{n\ge 0},\ t \in [0,1]$ is analyzed as well.

math.CA↗
arXiv2021

Discrete orthogonal polynomials associated with Macdonald function

New sequences of discrete orthogonal polynomials associated with the modified Bessel function $K_μ(z)$ or Macdonald function are considered. The corresponding weight function is $λ^k ρ_{k+ν+1}(t)/ k!$, where $\ k \in \mathbb{N}_0, \ t \ge 0,\ ν> -1,\ 0 < λ< 1,\ ρ_μ(z) = 2 z^{μ/2} K_μ\left( 2\sqrt z\right)$. The limit case $t=0$ corresponds to the Meixner polynomials. Various properties, differential-difference recurrence relations are established. The modified sequence of polynomials with the weight $λ^k ρ_{k+ν+1}(λt)/ k! $ is investigated as well.

math.CA↗
arXiv2021

Orthogonal polynomials for the weight $x^ν \exp(-x - t/x)$

Orthogonal polynomials for the weight $x^ν \exp(-x - t/x),\ x, t > 0, ν\in \mathbb{R}$ are investigated. Differential-difference equations, recurrence relations, explicit representations, generating functions and Rodrigues-type formula are obtained.

math.CA↗
arXiv2021

A method of composition orthogonality and new sequences of orthogonal polynomials and functions for non-classical weights

A new method of composition orthogonality is introduced. It is applied to generate new sequences of orthogonal polynomials and functions. In particular, classical orthogonal polynomials are interpreted in the sense of composition orthogonality. Finally, new sequences of orthogonal polynomials with respect to the weight function $x^αρ^2_ν(x), ρ_ν(x)= 2 x^{ν/2} K_ν(2\sqrt x),\ x >0, ν\ge 0, α> -1$, where $K_ν(z)$ is the modified Bessel function or Macdonald function, are investigated. Differential properties, recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are obtained. The corresponding multiple orthogonal polynomials are exhibited.

math.CA↗