arXiv subjects
Publications and source records attributed to S. M. Sitnik.
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 preprint we consider generalizations of discrete and integral Cauchy--Bunyakovskii inequalities by the method of mean values with some applications. Mostly the material is compiled as a short survey but some results are proved. Main results are listed, including an interesting inequality with maximum and minimum values. Some applications are considered from different fields of mathematics. Among them are estimates for some special functions, including Euler gamma and incomplete gamma function, the Legendre complete elliptic integrals of the first kind. Also some further possible generalizations are considered and outlined, including generalizations of the Aczél and Minkovskii inequalities, a case of spaces with sign-indefinite form, the Jackson's $q$-integrals.
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 consider interpolation problem connected with series by integer shifts of Gaussians. Known approaches for these problems met numerical difficulties. Due to it another method is considered based on finite-rank approximations by linear systems. The main result for this approach is to establish correctness of the finite-rank linear system under consideration. And the main result of the paper is to prove correctness of the finite-rank linear system approximation. For that an explicit formula for the main determinant of the linear system is derived to demonstrate that it is non-zero.
The main content of this book is composed from two doctoral theses: by V.\,V.~Katrakhov (1989) and by S.\,M.~Sitnik (2016). In our work, for the first time in the format of a monograph, we systematically expound the theory of transmutation operators and their applications to differential equations with singularities in coefficients, in particular, with Bessel operators. Along with detailed survey and bibliography on this theory, the book contains original results of the authors. Significant part of these results is published with detailed proofs for the first time. The book is concluded with a brief biographic essay about Valeriy V.~Katrakhov, as well as detailed bibliography containing 648 references.
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. This short survey paper contains brief historical information, main known facts and original author's results on the theory of the Buschman-Erdélyi transmutations and some of their applications. The operators of Buschman-Erdélyi type were first studied by E.T.~Copson, R.G.~Buschman and A.~Erdélyi as integral operators. In 1990's the author was first to prove the transmutational nature of these operators and published papers with detailed study of their properties. This class include as special cases such famous objects as the Sonine-Poisson-Delsarte transmutations and the fractional Riemann-Lioville integrals. In this paper, the Buschman-Erdelyi transmutations are logically classified as operators of the first kind with special case of zero order smoothness operators, second kind and third kind with special case of unitary Sonine-Katrakhov and Poisson-Katrakhov transmutations. We study such properties as transmutational conditions, factorizations, norm estimates, connections with classical integral transforms. Applications are considered to singular partial differential equations, embedding theorems with sharp constants in Kipriyanov spaces, Euler-Poisson-Darboux equation including Copson lemma, generalized translations, Dunkl operators, Radon transform, generalized harmonics theory, Hardy operators, V.\,Katrakhov's results on pseudodifferential operators and boundary-value problems of new kind for equations with solutions of arbitrary growth at the isolated singularity for elliptic partial differential equations.
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.
This is a preprint of 1992 with some updates. We study sections of the exponential function Taylor series. Interesting inequalities for these sections were considered by G.Hardy, Kesava Menon, W. Gautschi, H.Alzer and others. The main aim of this preprint is to investigate new proofs for the main inequality with best constants and its multiple generalizations. Some conjectures are formulated (and some of them were proved recently, see comments of 2016).
We investigate inequalities of M.G. Krein, Yu.V. Linnik and E.A. Gorin for positive definite functions. Modifications and generalizations of these inequalities are proved. We also prove that multipoint E.A. Gorin's inequality follows from two--point M.G. Krein's inequality.
We consider Gini means with short biographical information and propose a new proof of the main inequality for these means. Also some applications of Gini and other means are considered to polymer chemistry.
We consider a new and simpler proof of an inequality of A.S. Gasparyan, which was originally derived in terms of complex algebraical objects --- multidimensional hyperdeterminants. Our proof is much simpler and use only standard technics such as mean inequalities. The main theorem on Gini means is essentially used. Some corollaries and problems are considered.
In 2016 Prof. Fozi M. Dannan from Damascus, Syria proposed an inequality for three positive numbers with unit product. It became widely known but was not proved yet in spite of elementary formulation. We give some proofs for this inequality, a number of its generalizations and some connected unsolved problems and conjectures.
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.
In the preprint of 1993 the author formulated some conjectures on monotonicity of ratios for exponential series remainders. They are equivalent to conjectures on monotonicity of a ratio of Kummer hypergeometric functions and presumably not proved still. In this short note the most interesting conjecture from the preprint of 1993 is reproduced with its generalizations. In the updated version an important information is added that conjectures 1, 2 and also problems 1, 2 were recently proved by Khaled Mehrez from IPEIM, Tunisia, for details cf. arXiv:1410.6120. Also the results on monotonicity were generalized to basic hypergeometric functions.
In this paper we study single-parametric systems of integer shifts of Gauss and Lorenz functions. In case of Cauchy--Lorenz system we explicitly calculate nod functions and prove that it tends to sinc function in limit. For both Gauss and Cauchy--Lorenz systems and corresponding nod functions we explicitly calculate Riesz constants via trigonometric, hyperbolic and Jacobi theta-functions, also limit behavior of this values is found depending on parameters. A special result is a sharp monotonicity property proved for a special ratio of Jacobi theta-functions which is important in many areas. Brief discussion of numerical methods is included with additional references. Different applications are outlined to coherent states and atomic spectra. Some unsolved problems and comments are added.
We prove that the main theorem for absolutely monotonic functions from the book Mitrinović~D.S., Pečarić~J.E., Fink~A.M. "Classical And New Inequalities In Analysis", Kluwer Academic Publishers, 1993, is not valid without additional restrictions.
We find sufficient conditions for log-convexity and log-concavity for the functions of the forms $a\mapsto\sum{f_k}(a)_kx^k$, $a\mapsto\sum{f_k}Γ(a+k)x^k$ and $a\mapsto\sum{f_k}x^k/(a)_k$. The most useful examples of such functions are generalized hypergeometric functions. In particular, we generalize the Turán inequality for the confluent hypergeometric function recently proved by Barnard, Gordy and Richards and log-convexity results for the same function recently proved by Baricz. Besides, we establish a reverse inequality which complements naturally the inequality of Barnard, Gordy and Richards. Similar results are established for the Gauss and the generalized hypergeometric functions. A conjecture about monotonicity of a quotient of products of confluent hypergeometric functions is made.