Yu. A. Brychkov

Bio: Yu. A. Brychkov is an academic researcher from Russian Academy of Sciences. The author has contributed to research in topics: Special functions & Hypergeometric function. The author has an hindex of 11, co-authored 47 publications receiving 4240 citations.

Integrals and Series

Abstract: The pages of this expensive but invaluable reference work are dense with formulae of stupefying complexity. Chapters 1 and 2 treat definite/indefinite integral properties of a great variety of special functions, Chapters 3 and 4 (which are relatively brief) treat definite integrals of some piece-wi

Integral transforms of generalized functions

Abstract: One surveys the investigations on the integral transforms of generalized functions Basically, the survey convers the papers and monographs published after 1974

On some formulas for the Appell function F3 (a,a′, b, b′; c; w,z)

Abstract: New relations and transformation formulas for the Appell function F3(a,a′,b,b′;c;w;z) and the confluent Appell functions (Humbert functions) Φ2,Φ3,Ξ1,Ξ2 are obtained. These relations include limit formulas, integral representations, differentiation formulas. Various finite and infinite summation formulas are also derived.

On some properties of the Marcum Q function

Abstract: A closed expression for Q ν(a, b) with integer ν in terms of a confluent Appell function, differentiation formulas with respect to a and b, generating functions and other relations are given.

Integrals and Series : Direct Laplace Transforms

Abstract: Volumes 4 and 5 of the extensive series Integrals and Series are devoted to tables of LaplaceTransforms In these companion volumes the authors have collected data scatteredthroughout the literature, and have augmented this material with many unpublished resultsobtained in their own researchVolume 4 contains tables of direct Laplace transforms, a number of which are expressed interms of the Meijer G-function When combined with the table of special cases, theseformulas can be used to obtain Laplace transforms of numerous elementary and specialfunctions of mathematical physicsVolume 5 offers tables of inversion formulas for the Laplace transformation and includestables of factorization and inversion of various integral transforms

The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system

Abstract: The most voluminous bibliography of the analytical methods for calculating of integrals is represented in the article [19]. It is shown there that the most effective and the simplest algorithm of analytical integration was made by O.I. Marichev [8, 9, 12]. Later it was realized in the reference-books [16-18, 20]. This algorithm allows us to calculate definite and indefinite integrals of the products of elementary and special functions of hypergeometric type. It embraces about 70 per cent of integrals which are included in the world reference-literature. It allows to calculate many other integrals too.The present article contains short description of this algorithm and its realization in the REDUCE system during the process of creation of INTEGRATOR system. Only one general method of integration is known to be realized on the computers, i.e. criterion algorithm for calculating of indefinite integrals of elementary functions through elementary functions by themselves (the authors of it are M. Bronstein and other).The idea of our algorithm is in the following. The initial integrals is transformed to contour integral from the ratio of products of gamma-functions by means of Mellin transform and parseval equality. The residue theorem is used for the calculating of the received integral which due to the strict rules results in sums of hypergeometric series. The value of integral itself and the integrand functions are the special cases of the well-known Meijer's G-function [4, 7, 8, 12, 14, 18].Programming packet is realized in programming languages PASCAL and REDUCE. It also offers the opportunity of finding the values for some classical integral transforms (Laplace, Hankel, Fourier, Mellin and etc.). The REDUCE's part of packet contains the main properties of the well-known special functions, such as the Bessel and gamma-functions and kindred functions, Anger function, Weber function, Whittaker functions, generalized hypergeometric functions. Special place in the packet is occupied by Meijers's G-function for which the main properties such as finding the particular cases and representation by means of hypergeometric series are realized.

Statistical Size Distributions in Economics and Actuarial Sciences

Abstract: A comprehensive account of economic size distributions around the world and throughout the years In the course of the past 100 years, economists and applied statisticians have developed a remarkably diverse variety of income distribution models, yet no single resource convincingly accounts for all of these models, analyzing their strengths and weaknesses, similarities and differences. Statistical Size Distributions in Economics and Actuarial Sciences is the first collection to systematically investigate a wide variety of parametric models that deal with income, wealth, and related notions. Christian Kleiber and Samuel Kotz survey, compliment, compare, and unify all of the disparate models of income distribution, highlighting at times a lack of coordination between them that can result in unnecessary duplication. Considering models from eight languages and all continents, the authors discuss the social and economic implications of each as well as distributions of size of loss in actuarial applications. Specific models covered include: Pareto distributions Lognormal distributions Gamma-type size distributions Beta-type size distributions Miscellaneous size distributions Three appendices provide brief biographies of some of the leading players along with the basic properties of each of the distributions. Actuaries, economists, market researchers, social scientists, and physicists interested in econophysics will find Statistical Size Distributions in Economics and Actuarial Sciences to be a truly one-of-a-kind addition to the professional literature.

The fundamental solution of the space-time fractional diffusion equation

Abstract: We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order � ∈ (0,2] and skewness � (|�| ≤ min {�,2 − �}), and the first-order time derivative with a Caputo derivative of order � ∈ (0,2]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We review the particular cases of space-fractional diffusion {0 < � ≤ 2, � = 1}, time-fractional diffusion {� = 2, 0 < � ≤ 2}, and neutral-fractional diffusion {0 < � = � ≤ 2}, for which the fundamental solution can be interpreted as a spatial probability density function evolving

Capacity of multiple-antenna fading channels: spatial fading correlation, double scattering, and keyhole

Abstract: The capacity of multiple-input multiple-output (MIMO) wireless channels is limited by both the spatial fading correlation and rank deficiency of the channel. While spatial fading correlation reduces the diversity gains, rank deficiency due to double scattering or keyhole effects decreases the spatial multiplexing gains of multiple-antenna channels. In this paper, taking into account realistic propagation environments in the presence of spatial fading correlation, double scattering, and keyhole effects, we analyze the ergodic (or mean) MIMO capacity for an arbitrary finite number of transmit and receive antennas. We assume that the channel is unknown at the transmitter and perfectly known at the receiver so that equal power is allocated to each of the transmit antennas. Using some statistical properties of complex random matrices such as Gaussian matrices, Wishart (1928) matrices, and quadratic forms in the Gaussian matrix, we present a closed-form expression for the ergodic capacity of independent Rayleigh-fading MIMO channels and a tight upper bound for spatially correlated/double scattering MIMO channels. We also derive a closed-form capacity formula for keyhole MIMO channels. This analytic formula explicitly shows that the use of multiple antennas in keyhole channels only offers the diversity advantage, but provides no spatial multiplexing gains. Numerical results demonstrate the accuracy of our analytical expressions and the tightness of upper bounds.

Mittag-Leffler Functions and Their Applications

Abstract: Motivated essentially by the success of the applications of the Mittag-Leffler functions in many areas of science and engineering, the authors present, in a unified manner, a detailed account or rather a brief survey of the Mittag-Leffler function, generalized Mittag-Leffler functions, Mittag-Leffler type functions, and their interesting and useful properties. Applications of G. M. Mittag-Leffler functions in certain areas of physical and applied sciences are also demonstrated. During the last two decades this function has come into prominence after about nine decades of its discovery by a Swedish Mathematician Mittag-Leffler, due to the vast potential of its applications in solving the problems of physical, biological, engineering, and earth sciences, and so forth. In this survey paper, nearly all types of Mittag-Leffler type functions existing in the literature are presented. An attempt is made to present nearly an exhaustive list of references concerning the Mittag-Leffler functions to make the reader familiar with the present trend of research in Mittag-Leffler type functions and their applications.