Multiple hypergeometric series -- Appell series and beyond

TL;DR

This survey consolidates core aspects of Appell-type multivariate hypergeometric series, detailing definitions, recurrences, PDEs, integral representations, and transformations for the four two-variable functions $F_1$–$F_4$ and their connections to more general families. It situates Appell functions within a broader landscape that includes Horn, Kampé de Fériet, and Lauricella series, highlighting integral representations and reduction identities that enable practical manipulations and applications, including links to quantum field theory and Feynman integrals. The work also notes deeper structural interpretations, such as Lie-algebraic symmetries and group representations, and points to extensions like GKZ $A$-hypergeometric functions as a further horizon beyond the scope. Overall, it provides a compact, technique-oriented toolkit for working with Appell-type series and their natural generalizations.

Abstract

This survey article (which will appear as a chapter in the book ``Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions'', Springer-Verlag) provides a small collection of basic material on multiple hypergeometric series of Appell-type and of more general series of related type.