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2014 | Buch

Kapitel lesen Erstes Kapitel lesen

Hypergeometric Summation

An Algorithmic Approach to Summation and Special Function Identities

verfasst von: Wolfram Koepf

Verlag: Springer London

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Modern algorithmic techniques for summation, most of which were introduced in the 1990s, are developed here and carefully implemented in the computer algebra system Maple™.

The algorithms of Fasenmyer, Gosper, Zeilberger, Petkovšek and van Hoeij for hypergeometric summation and recurrence equations, efficient multivariate summation as well as q-analogues of the above algorithms are covered. Similar algorithms concerning differential equations are considered. An equivalent theory of hyperexponential integration due to Almkvist and Zeilberger completes the book.

The combination of these results gives orthogonal polynomials and (hypergeometric and q-hypergeometric) special functions a solid algorithmic foundation. Hence, many examples from this very active field are given.

The materials covered are suitable for an introductory course on algorithmic summation and will appeal to students and researchers alike.

Inhaltsverzeichnis

Frontmatter

Chapter 1. The Gamma Function

Abstract

Apart from the elementary transcendental functions such as the exponential and trigonometric functions and their inverses, the Gamma function is probably the most important transcendental function. It was defined by Euler to interpolate the factorials at noninteger arguments.

Wolfram Koepf

Chapter 2. Hypergeometric Identities

Abstract

In this chapter we deal with hypergeometric identities.

Wolfram Koepf

Chapter 3. Hypergeometric Database

Abstract

In this chapter we list some of the major hypergeometric identities. Note that most of these do not require any variables to have integer values. We give examples showing how this database can be used to generate binomial identities.

Wolfram Koepf

Chapter 4. Holonomic Recurrence Equations

Abstract

The main algorithmic idea for finding hypergeometric term representations of hypergeometric series goes back to Celine Fasenmyer (often called Sister Celine): Her idea is to find a recurrence equation for the sum. If the resulting recurrence equation can be solved explicitly, you are done.

Wolfram Koepf

Chapter 5. Gosper’s Algorithm

Abstract

For a moment, let’s have a break from searching for hypergeometric term solutions and recurrence equations of infinite series. Instead, we will deal with sums with variable limits of summation, an interesting topic in itself. Later, this will prove to be a useful tool in discovering an algorithmic method for infinite sums.

Wolfram Koepf

Chapter 6. The Wilf-Zeilberger Method

Abstract

In this chapter, we study the connection between Gosper’s algorithm and definite sums. Firstly, we give a direct application of Gosper’s algorithm to definite summation. The application of Gosper’s algorithm to a modified input can prove definite hypergeometric identities.

Wolfram Koepf

Chapter 7. Zeilberger’s Algorithm

Abstract

In this chapter, we introduce Zeilberger’s extension of Gosper’s algorithm, using which one can not only prove hypergeometric identities but also sum definite series in many cases, if they represent hypergeometric terms.

Wolfram Koepf

Chapter 8. Extensions of the Algorithms

Abstract

In this chapter, we extend Gosper’s, Wilf-Zeilberger’s and Zeilberger’s methods to accept rational-linear inputs rather than only integer-linear ones. For such an input \(a_{k+1}/a_k\) is not always rational, so that Gosper’s algorithm may not apply.

Wolfram Koepf

Chapter 9. Petkovšek’s and van Hoeij’s Algorithm

Abstract

We saw that in many cases Zeilberger’s algorithm obtains the holonomic recurrence equation of lowest order for a given definite sum \(s_n\). In particular, if the order of the resulting recurrence equation is one, or if the latter contains only two shifts \(s_n\) and \(s_{n+m}\) for some \(m\in {\mathbb {N}}\), then one finds a hypergeometric term representation for the sum under consideration using \(m\) initial values. In this chapter we give algorithms which find all hypergeometric term solutions of a holonomic recurrence equation.

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Chapter 10. Differential Equations for Sums

Abstract

In this chapter, as an interesting variation of Zeilberger’s method, we present an algorithm which generates holonomic differential equations rather than recurrence equations for definite sums of a certain type.

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Chapter 11. Hyperexponential Antiderivatives

Abstract

In this chapter, we consider a continuous counterpart of Gosper’s algorithm. The appropriate question is to find a hyperexponential term antiderivative G(x) of a given f(x) whenever one exists.

Wolfram Koepf

Chapter 12. Holonomic Equations for Integrals

Abstract

Now we are ready to consider definite integration of hyperexponential terms. If the corresponding indefinite integral is a hyperexponential term again, then the continuous Gosper algorithm applies, and definite integration is trivial.

Wolfram Koepf

Chapter 13. Rodrigues Formulas and Generating Functions

Abstract

In this chapter we use the algorithms of the preceding chapter to obtain holonomic equations for function families given by Rodrigues type formulas and generating functions.

Wolfram Koepf

Backmatter

Titel: Hypergeometric Summation
verfasst von: Wolfram Koepf
Verlag: Springer London
Electronic ISBN: 978-1-4471-6464-7
Print ISBN: 978-1-4471-6463-0
DOI: https://doi.org/10.1007/978-1-4471-6464-7