This book is all it says in the title, and indeed rather more. It is a collection of papers that grew out of a summer school course on integration, summation, and special functions in quantum field theory, run by the editors’ institutions, the Deutsches Elektronen-Synchrotron (DESY) and the Research Institute for Symbolic Computation (RISC) at the University of Linz. There is, therefore, quite a lot of material on integration, summation, and special functions that has relevance beyond quantum field theory.
The individual papers are quite diverse, and cannot easily be summarized. Probably the best example of the power of computer algebra in this area is given by the third paper, “Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order.” As recounted there, when the NIST Digital Library of Mathematical Functions (DLMF) [1] was being constructed to replace the venerable Abramowitz and Stegun book [2], it was discovered that some identities in the book [2] had no extant proofs, and indeed there was a note by Stegun listing some errors, but again no proofs. The editor in chief of the DLMF [1] appealed to the RISC team for help, and all the identities were further corrected and proved within two weeks, using the tools the RISC team and others had provided to Maple and Mathematica. Without this, the DLMF [1] would not have been able to include these unproven statements.
One key tool that keeps appearing is holonomic sequences (an, defined by a recurrence relation in terms of an-1, . . . , an-r with coefficients depending polynomially on n) or functions (y(x), defined by a differential equation in terms of y’(x), . . . , y(r)(x) with coefficients depending polynomially on x). Here, Kauers’ paper, “The Holonomic Toolkit,” is a valuable introduction.
Much of this book is not easy reading. I struggled for a few days with this statement on page 4, “All these functions are transcendental to the previous ones,” before e-mailing the author and being referred to Ostrowski’s work [3], which wasn’t cited. But 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.
