Victor S. Adamchik

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

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 Meijers 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 REDUCEs 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 Meijerss G-function for which the main properties such as finding the particular cases and representation by means of hypergeometric series are realized.

On Stirling numbers and Euler sums

Abstract In this paper, we propose another yet generalization of Stirling numbers of the first kind for noninteger values of their arguments. We discuss the analytic representations of Stirling numbers through harmonic numbers, the generalized hypergeometric function and the logarithmic beta integral. We present then infinite series involving Stirling numbers and demonstrate how they are related to Euler sums. Finally, we derive the closed form for the multiple zeta function ζ(p, 1,…, 1) for Re(p)>1.

Some Series of the Zeta and Related Functions

We propose and develop yet another approach to the problem of summation of series involving the Riemann Zeta function (s), the (Hurwitzs) generalized Zeta function (s; a), the Polygamma function (z) (p = 0; 1; 2; ), and the polylogarithmic function Lis(z). The key ingredients in our approach include certain known integral representations for (s) and (s; a). The method developed in this paper is illustrated by numerous examples of closed-form evaluations of series of the aforementioned types; the method developed in Section 2, in particular, has been implemented in Mathematica (Version 3.0). Many of the resulting summation formulas are believed to be new. 1991Mathematics Subject Classi cation. Primary 11M06, 11M35; Secondary 33B15.

Multiple Gamma and related functions

The authors give several new (and potentially useful) relationships between the multiple Gamma functions and other mathematical functions and constants. As by-products of some of these relationships, a classical definite integral due to Euler and other definite integrals are also considered together with closed-form evaluations of some series involving the Riemann and Hurwitz Zeta functions.

On the Barnes function

The multiple Barnes function, defined as a generalization of the Euler gamma function, is used in many applications of pure and applied mathematics and theoretical physics. This paper presents new integral representations as well as special values of the Barnes function. Moreover, the Barnes function is expressed in a closed form by means of the Hurwitz zeta function. These results can be used for numeric and symbolic computations of the Barnes function.

A class of logarithmic integrals

A class of de nite integrals involving cyclotomic polynomials and nested logarithms is considered. The results are given in terms of derivatives of the Hurwitz Zeta function. Some special cases for which such derivatives can be expressed in closed form are also considered. The integration procedure is implemented in Mathematica V3.1.

Universality of finite-size corrections to the number of critical percolation clusters

Monte-Carlo simulations on a variety of 2d percolating systems at criticality suggest that the excess number of clusters in finite systems over the bulk value of nc is a universal quantity, dependent upon the system shape but independent of the lattice and percolation type. Values of nc are found to high accuracy, and for bond percolation confirm the theoretical predictions of Temperley and Lieb, and Baxter, Temperley, and Ashley, which we have evaluated explicitly in terms of simple algebraic numbers. Predictions for the fluctuations are also verified for the first time.

The evaluation of integrals of Bessel functions via G -function identities

A few transformations are presented for reducing certain cases of Meijers G-function to a G-function of lower order. Their applications to the integration of a product of Bessel functions are given. The algorithm has been implemented within Mathematica 3.0.

Symbolic and Numeric Computations of the Barnes Function

This paper discusses some theoretical aspects and algorithms for high-precision computation of the Barnes gamma function.

A learning objects approach to teaching programming

The goal of this paper is to describe a new approach to a content creation and delivery mechanism for a programming course. This approach is based on the concept of creating a large repository of learning objects, each of which consists of the core material, code examples, supplementary notes, and review questions. A learning object will be uniquely described by a XML document and presents an interface for future search, retrieval and updating, as well as for potential connection to external assessment tools. Furthermore, we describe a new teaching, learning and authoring tool (called adaptive book) that allows users to add new learning objects, modify current ones, and discuss concepts using a variety of representation models. With the adaptive book, an instructor will be able to design his or her very own course using a large repository of material, which will target a particular audience or a customized syllabus. In addition, the electronic adaptive book will serve as an interactive, continuously up-to-date learning environment for students. It will allow students to create personal learning profiles that are embedded into the core content of the course. We believe that in the near future, this electronic adaptive book will have the potential to replace or enhance traditional paper textbooks. In all physicality, it is impossible to have a personal instructor for each student, but with the adaptive book, we believe we are one step closer to realizing the dream of individualized instructions for each and every student.