Special Functions

*To*: mathgroup at yoda.physics.unc.edu*Subject*: Special Functions*From*: ernst at net.neic.nsk.su (Krupnikov Ernst Davidovich)*Date*: Fri, 25 Jun 93 22:18:19 +0600

Dear colleagues: the fifth academic year I am training physicist students of the Novosibirsk University in generalized hypergeometric functions in one and two variables. This academic year I am finishing preparation of my exercise book "Computer-Oriented Algorithms for Special Functions of Mathematical Physics or Rewriting Rules for Generalized Hypergeometric Functions" (in Russian; about 100 pages without solutions; typed by ChiWriter 3.14). I would like to be invited for a term or two during the next academic year (1993-94) for training your under- and post-graduates in this subject and simultaneous preparation of a revised and enlarged version of this educational book. I am ready to answer your questions and to exchange my know how for somebody's invitation. Ernst Davidovich Krupnikov E-mail: ernst at net.neic.nsk.su Postal address: R E G I S T E R E D Krupnikov E.D. P.O.Box 300 Novosibirsk 90 630090, Russian Federation Table of Contents Preface Introduction Literature for self-education and reference books Chapter 1.Generalized hypergeometric function of one variable 1.1.Gamma function and Pochhammer symbol 1.2.Canonical form for Pochhammer symbol 1.3.Definition of generalized hypergeometric function of one variable. Direct and inverse dictionaries 1.4.Saalschutz, Vandermonde, Gauss summation theorems and their interconnections 1.5.Interconnections between four summations theorems for F 2 1 1.6.Summation theorems for F and interconnections between them 3 2 1.7.Calculation of sums of rational fractions 1.8.Calculation of combinatorial sums 1.9.Splitting up sums into odd and even terms 1.10.Reversion of summation 1.11.Reduction identities for generalized hypergeometric function 1.12.Some useful limits 1.13.Three-term relations for generalized hypergeometric function 1.14.Addition tables for F , F , F , F 0 1 1 1 2 1 3 2 1.15.Generalization of summation theorems 1.16.Logarithmic derivative of gamma function and Riemann zeta function of integer argument Chapter 2.Kampe de Feriet function 2.1.Definition and basic properties of KdF function 2.2.Properties of KdF function as corollaries of properties of concrete hypergeometric functions of one variable (beginning) 2.3.Appell and Humbert functions as special cases of KdF function 2.4.Multiplication formula for gamma function 2.5.Definite integrals of special functions 2.6.Properties of KdF function as corollaries of summation theorems for generalized hypergeometric functions of one variable (continuation of 2.2) 2.6a.Isolation of real and imaginary parts of hypergeometric functions 2.7.Splitting up KdF function into more simple functions 2.8.Differentiation of Pochhammer symbol with respect to parameter 2.9.Differentiation of generalized hypergeometric function of one variable with respect to parameter 2.10.Reduction of multiplicity of sums Parting words