arXiv subjects
Publications and source records attributed to E. L. Shishkina.
This paper presents a study of the Weber-Schafheitlin integral and its connections to random walk theory. The Weber-Schafheitlin integral, defined as the improper integral of a product of Bessel functions, arises naturally in the evaluation of probability distributions for multidimensional random walks. This work bridges pure special function theory with applied probability.
In this paper, we introduce the new construction of fractional derivatives and integrals with respect to a function, based on a matrix approach. We believe that this is a powerful tool in both analytical and numerical calculations. We begin with the differential operator with respect to a function that generates a semigroup. By discretizing this operator, we obtain a matrix approximation. Importantly, this discretization provides not only an approximating operator but also an approximating semigroup. This point motivates our approach, as we then apply Balakrishnan's representations of fractional powers of operators, which are based on semigroups. Using estimates of the semigroup norm and the norm of the difference between the operator and its matrix approximation, we derive the convergence rate for the approximation of the fractional power of operators with the fractional power of correspondings matrix operators. In addition, an explicit formula for calculating an arbitrary power of a two-band matrix is obtained, which is indispensable in the numerical solution of fractional differential and integral equations.
September 7, 2025 marked the 80th anniversary of the birth of Oleg Marichev. Marichev is known mathematician which has developed many of Mathematica's algorithms for the calculation of definite and indefinite integrals and hypergeometric functions including Meijer G-function.
In this article we would like to consider some approaches to non-integer integro-differentiations and its implementation in computer algebra system Wolfram Mathematics.
This is a survey paper in two parts. In the first part we list main variants of one-dimensional fractional integrodifferential operators. Also some historical and priority remarks are given. As a special question we consider the impact of Soviet and Russian researches to fractional calculus and its applications to viscoelasticity theory. We also stress an impact of the Voronezh school of mechanics and viscoelasticity theory, including works of Shermergor, Meschkov, Rossikhin, Shitikova and others. In the second part of the paper we consider two important special classes of generalized fractional operators, they were thoroughly studied by the authors. We consider Buschman-Erdelyi operators, this is an important class containing as special cases Riemann-Liouville, Erdélyi-Kober, Mehler-Fock operators and some others. They also include classical transmutations for the Bessel differential operator, namely Sonine and Poisson ones. After that we represent results on another important generalization of Riemann-Liouville operators, namely fractional powers of the Bessel operators. The paper is dedicated to Adam Maremovich Nakhushev -- brilliant man and mathematician.
In this paper we study fractional powers of the Bessel differential operator defined on a semiaxis. Some important properties of such fractional powers of the Bessel differential operator are proved. They include connections with Legendre functions for kernel representations, fractional integral operators of Liouville and Saigo, Mellin transform and index laws. Possible applications are indicated to differential equations with fractional powers of the Bessel differential operator.
This paper was published in the special issue of the Journal of Inequalities and Special Functions dedicated to Professor Ivan Dimovski's contributions to different fields of mathematics: transmutation theory, special functions, integral transforms, function theory etc. In this paper we study fractional powers of the Bessel differential operator. The fractional powers are defined explicitly in the integral form without use of integral transforms in its definitions. Some general properties of the fractional powers of the Bessel differential operator are proved and some are listed. Among them are different variations of definitions, relations with the Mellin and Hankel transforms, group property, generalized Taylor formula with Bessel operators, evaluation of resolvent integral operator in terms of the Wright or generalized Mittag--Leffler functions. At the end, some topics are indicated for further study and possible generalizations. Also the aim of the paper is to attract attention and give references to not widely known results on fractional powers of the Bessel differential operator.
In this article we consider certain types of weighted generalized functions associated with nondegenerate quadratic forms. Such functions and their derivatives are used for constructing fundamental solutions of iterated ultra-hyperbolic equations with Bessel operator and for constructing negative real powers of ultra-hyperbolic operators with Bessel operator.
Spherical means are well-known useful tool in the theory of partial differential equations with applications to solving hyperbolic and ultrahyperbolic equations and problems of integral geometry, tomography and Radon transforms. We generalize iterated spherical means to weighted ones based on generalized translation operators and consider applications to B-hyperbolic equations and transmission tomography problems.