main-content

## Über dieses Buch

The book focuses on advanced computer algebra methods and special functions that have striking applications in the context of quantum field theory. It presents the state of the art and new methods for (infinite) multiple sums, multiple integrals, in particular Feynman integrals, difference and differential equations in the format of survey articles. The presented techniques emerge from interdisciplinary fields: mathematics, computer science and theoretical physics; the articles are written by mathematicians and physicists with the goal that both groups can learn from the other field, including most recent developments. Besides that, the collection of articles also serves as an up-to-date handbook of available algorithms/software that are commonly used or might be useful in the fields of mathematics, physics or other sciences.

## Inhaltsverzeichnis

### Harmonic Sums, Polylogarithms,Special Numbers, and Their Generalizations

Abstract
In these introductory lectures we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and massive Feynman diagrams at higher loops. These quantities are elements of stuffle and shuffle algebras implying algebraic relations being widely independent of the special quantities considered. They are supplemented by structural relations. The generalizations are given in terms of generalized harmonic sums, (generalized) cyclotomic sums, and sums containing in addition binomial and inverse-binomial weights. To all these quantities iterated integrals and special numbers are associated. We also discuss the analytic continuation of nested sums of different kind to complex values of the external summation bound N.
Jakob Ablinger, Johannes Blümlein

### Multiple Zeta Values and Modular Forms in Quantum Field Theory

Abstract
This article introduces multiple zeta values and alternating Euler sums, exposing some of the rich mathematical structure of these objects and indicating situations where they arise in quantum field theory. Then it considers massive Feynman diagrams whose evaluations yield polylogarithms of the sixth root of unity, products of elliptic integrals, and L-functions of modular forms inside their critical strips.

### Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order

Abstract
We employ computer algebra algorithms to prove a collection of identities involving Bessel functions with half-integer orders and other special functions. These identities appear in the famous Handbook of Mathematical Functions, as well as in its successor, the DLMF, but their proofs were lost. We use generating functions and symbolic summation techniques to produce new proofs for them.
Stefan Gerhold, Manuel Kauers, Christoph Koutschan, Peter Paule, Carsten Schneider, Burkhard Zimmermann

### Conformal Methods for Massless Feynman Integrals and Large N f Methods

Abstract
We review the large N method of calculating high order information on the renormalization group functions in a quantum field theory which is based on conformal integration methods. As an example these techniques are applied to a typical graph contributing to the β-function of O(N) ϕ4 theory at O(1∕N 2). The possible future directions for the large N methods are discussed in light of the development of more recent techniques such as the Laporta algorithm.
John A. Gracey

### The Holonomic Toolkit

Abstract
This is an overview over standard techniques for holonomic functions, written for readers who are new to the subject. We state the definition for holonomy in a couple of different ways, including some concrete special cases as well as a more abstract and more general version. We give a collection of standard examples and state several fundamental properties of holonomic objects. Two techniques which are most useful in applications are explained in some more detail: closure properties, which can be used to prove identities among holonomic functions, and guessing, which can be used to generate plausible conjectures for equations satisfied by a given function.
Manuel Kauers

### Orthogonal Polynomials

Abstract
This chapter gives a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.
Tom H. Koornwinder

### Creative Telescoping for Holonomic Functions

Abstract
The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of a survey article: the developments in this area during the last two decades are sketched and a selection of references is compiled in order to highlight the impact of creative telescoping in numerous contexts.
Christoph Koutschan

### Renormalization and Mellin Transforms

Abstract
We study renormalization in a kinetic scheme (realized by subtraction at fixed external parameters as implemented in the BPHZ and MOM schemes) using the Hopf algebraic framework, first summarizing and recovering known results in this setting. Then we give a direct combinatorial description of renormalized amplitudes in terms of Mellin transform coefficients, featuring the universal property of rooted trees H R . In particular, a special class of automorphisms of H R emerges from the action of changing Mellin transforms on the Hochschild cohomology of perturbation series. Furthermore, we show how the Hopf algebra of polynomials carries a refined renormalization group property, implying its coarser form on the level of correlation functions. Application to scalar quantum field theory reveals the scaling behaviour of individual Feynman graphs.
Dirk Kreimer, Erik Panzer

### Relativistic Coulomb Integrals and Zeilberger’s Holonomic Systems Approach. I

Abstract
With the help of computer algebra we study the diagonal matrix elements ⟨Or p ⟩, where O $$= \left \{1,\beta,i\boldsymbol{\alpha }\mathbf{n}\beta \right \}$$ are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem. Using Zeilberger’s extension of Gosper’s algorithm and a variant to it, three-term recurrence relations for each of these expectation values are derived together with some transformation formulas for the corresponding generalized hypergeometric series. In addition, the virial recurrence relations for these integrals are also found and proved algorithmically.
Peter Paule, Sergei K. Suslov

### Hypergeometric Functions in Mathematica ®

Abstract
This paper is a short introduction to the generalized hypergeometric functions, with some theory, examples and notes on the implementation in the computer algebra system Mathematica ®. (Mathematica is a registered trademark of Wolfram Research, Inc.)
Oleksandr Pavlyk

### Solving Linear Recurrence Equations with Polynomial Coefficients

Abstract
Summation is closely related to solving linear recurrence equations, since an indefinite sum satisfies a first-order linear recurrence with constant coefficients, and a definite proper-hypergeometric sum satisfies a linear recurrence with polynomial coefficients. Conversely, d’Alembertian solutions of linear recurrences can be expressed as nested indefinite sums with hypergeometric summands. We sketch the simplest algorithms for finding polynomial, rational, hypergeometric, d’Alembertian, and Liouvillian solutions of linear recurrences with polynomial coefficients, and refer to the relevant literature for state-of-the-art algorithms for these tasks. We outline an algorithm for finding the minimal annihilator of a given P-recursive sequence, prove the salient closure properties of d’Alembertian sequences, and present an alternative proof of a recent result of Reutenauer’s that Liouvillian sequences are precisely the interlacings of d’Alembertian ones.
Marko Petkovšek, Helena Zakrajšek

### Generalization of Risch’s Algorithm to Special Functions

Abstract
Symbolic integration deals with the evaluation of integrals in closed form. We present an overview of Risch’s algorithm including recent developments. The algorithms discussed are suited for both indefinite and definite integration. They can also be used to compute linear relations among integrals and to find identities for special functions given by parameter integrals. The aim of this presentation is twofold: to introduce the reader to some basic ideas of differential algebra in the context of integration and to raise awareness in the physics community of computer algebra algorithms for indefinite and definite integration.
Clemens G. Raab

### Multiple Hypergeometric Series: Appell Series and Beyond

Abstract
This survey article provides a small collection of basic material on multiple hypergeometric series of Appell-type and of more general series of related type.
Michael J. Schlosser

### Simplifying Multiple Sums in Difference Fields

Abstract
In this survey article we present difference field algorithms for symbolic summation. Special emphasize is put on new aspects in how the summation problems are rephrased in terms of difference fields, how the problems are solved there, and how the derived results in the given difference field can be reinterpreted as solutions of the input problem. The algorithms are illustrated with the Mathematica package Sigma by discovering and proving new harmonic number identities extending those from Paule and Schneider, 2003. In addition, the newly developed package EvaluateMultiSums is introduced that combines the presented tools. In this way, large scale summation problems for the evaluation of Feynman diagrams in QCD (Quantum ChromoDynamics) can be solved completely automatically.
Carsten Schneider

### Potential of FORM 4.0

Abstract
I describe the main new features of Form version 4.0. They include factorization, polynomial arithmetic, new special functions, systems independent. sav files, a complete ParForm, open source code and a forum for user communication. I also mention a completely new feature for code simplification.
Jos A. M. Vermaseren

### Feynman Graphs

Abstract
In these lectures I discuss Feynman graphs and the associated Feynman integrals. Of particular interest are the classes functions, which appear in the evaluation of Feynman integrals. The most prominent class of functions is given by multiple polylogarithms. The algebraic properties of multiple polylogarithms are reviewed in the second part of these lectures. The final part of these lectures is devoted to Feynman integrals, which cannot be expressed in terms of multiple polylogarithms. Methods from algebraic geometry provide tools to tackle these integrals.
Stefan Weinzierl

### Backmatter

Weitere Informationen