Series expansions for the third incomplete elliptic integral via partial fraction decompositions

We find convergent double series expansions for Legendre's third incomplete elliptic integral valid in overlapping subdomains of the unit square. Truncated expansions provide asymptotic approximations in the neighbourhood of the logarithmic singularity $(1,1)$ if one of the variables approaches this point faster than the other. Each approximation is accompanied by an error bound. For a curve with an arbitrary slope at $(1,1)$ our expansions can be rearranged into asymptotic expansions depending on a point on the curve. For reader's convenience we give some numeric examples and explicit expressions for low-order approximations.Comment: The paper has been substantially updated (hopefully improved) and divided in two parts. This part is about third incomplete elliptic integral. 10 page

Relativistic Description of Exclusive Deuteron Breakup Reactions

The exclusive deuteron break-up reaction is analyzed within a covariant approach based on the Bethe-Salpeter equation with realistic meson-exchange interaction. Relativistic effects in the cross section, tensor analyzing power and polarization transfer are investigated in explicit form. Results of numerical calculations are presented for kinematical conditions in forthcoming p + D reactions at COSY.Comment: 10 LaTeX pages, 4 eps-figure

A SHORT SURVEY OF RECENT RESULTS ON BUSCHMAN-ERDELYI TRANSMUTATIONS

This short survey paper contains brief historical information, main known facts and original author's results on the theory of the Buschman-Erdelyi transmutations and some of their applications. The operators of Buschman-Erdelyi type were fi rst studied by E. T. Copson, R. G. Buschman and A. Erdelyi as integral operators. In 1990' s the author was fi rst 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 classi fi ed as operators of the fi rst 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 di ff erential 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 pseudodi ff erential operators and boundary-value problems of new kind for equations with solutions of arbitrary growth at the isolated singularity for elliptic partial di ff erential equations

A SHORT SURVEY OF RECENT RESULTS ON BUSCHMAN-ERDELYI TRANSMUTATIONS

This short survey paper contains brief historical information, main known facts and original author's results on the theory of the Buschman-Erdelyi transmutations and some of their applications. The operators of Buschman-Erdelyi type were fi rst studied by E. T. Copson, R. G. Buschman and A. Erdelyi as integral operators. In 1990' s the author was fi rst 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 classi fi ed as operators of the fi rst 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 di ff erential 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 pseudodi ff erential operators and boundary-value problems of new kind for equations with solutions of arbitrary growth at the isolated singularity for elliptic partial di ff erential equations

On an identity for the iterated weighted spherical mean and its applications

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. © 2016 Shishkina, E.L., Sitnik, S.M

On monotonicity of ratios of some hypergeometric functions

In the preprint [1] one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. They lead to more general conjecture on monotonicity of ratios of Kummer hypergeometric functions and were not proved from 1993. In this paper we prove some conjectures from [1] for Kummer hypergeometric functions and its further generalizations for Gauss and generalized hypergeometric functions. The results are also closely connected with Turán-type inequalities for special functions

On monotonicity of ratios of some q-hypergeometric functions

In this paper we prove monotonicity of some ratios of q-Kummer confluent hypergeometric and q-hypergeometric functions. The results are also closely connected with Turán type inequalities. In order to obtain main results we apply methods developed for the case of classical Kummer and Gauss hypergeometric functions in [S.M. Sitnik, Inequalities for the exponential remainder, preprint, Institute of Automation and Control Process, Far Eastern Branch of the Russian Academy of Sciences, Vladivostok 1993 (in Russian)] and [S.M. Sitnik, Conjectures on Monotonicity of Ratios of Kummer and Gauss Hypergeometric Functions, RGMIA Research Report Collection 17 (2014), Article 107]. © 2016, Drustvo Matematicara Srbije. All rights reserved

General form of the Euler-Poisson-Darboux equation and application of the transmutation method

In this article, we find solution representations in the compact integral form to the Cauchy problem for a general form of the Euler-Poisson-Darboux equation with Bessel operators via generalized translation and spherical mean operators for all values of the parameter k, including also not studying before exceptional odd negative values. We use a Hankel transform method to prove results in a unified way. Under additional conditions we prove that a distributional solution is a classical one too. A transmutation property for connected generalized spherical mean is proved and importance of applying transmutation methods for differential equations with Bessel operators is emphasized. The paper also contains a short historical introduction on differential equations with Bessel operators and a rather detailed reference list of monographs and papers on mathematical theory and applications of this class of differential equations. © 2017 Texas State University

Functional Inequalities for the Mittag–Leffler Functions

In this paper, some Turán-type inequalities for Mittag–Leffler functions are considered. The method is based on proving monotonicity for special ratio of sections for series of Mittag–Leffler functions. Furthermore, we deduce the Lazarević- and Wilker-type inequalities for Mittag–Leffler functions. © 2017, Springer International Publishing