MathGroup Archive 1993

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Special Functions

  • To: mathgroup at
  • Subject: Special Functions
  • From: ernst at (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
          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
 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
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

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