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This book focuses on advanced computer algebra methods and special functions that have applications in quantum field theory. It presents new methods for (infinite) multiple sums, multiple integrals, difference and differential equations.

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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.

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Harmonic sums, polylogarithms, special numbers, and their generalizations.- Multiple Zeta values and modular forms in quantum field theory.- Computer-assisted proofs of some identities for Bessel functions of fractional order.- Conformal methods for massless Feynman integrals and large Nf methods.- The holonomic toolkit.- Orthogonal polynomials.- Creative telescoping for holonomic functions.- Renormalization and Mellin transforms.- Relativistic Coulomb integrals and Zeilberger´s holonomic systems approach.- Hypergeometric functions in Mathematica.- Solving linear recurrence equations with polynomial coefficients.- Generalization of Risch´s algorithms to special functions.- Multiple hypergeometric series.- Appell series and beyond.- Simplifying multiple sums in difference fields.- Potential of FORM 4.0.- Feynman graphs.

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From the reviews:

"It is a collection of papers that grew out of a summer school course on integration, summation, and special functions in quantum field theory ... . if you are interested in these sorts of special functions, and the computer algebra tools to manipulate them, whether or not your particular application is quantum field theory, then this book is an excellent description of the state of the art in computer algebra manipulation and proof." (J. H. Davenport, Computing Reviews, April, 2014)

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