Hypergeometric Functions and Feynman Diagrams

  title={Hypergeometric Functions and Feynman Diagrams},
  author={Mikhail Yu. Kalmykov and Vladimir V. Bytev and Bernd A. Kniehl and S. Moch and Bennie F. L. Ward and Scott A. Yost},
  journal={Texts \& Monographs in Symbolic Computation},
The relationship between Feynman diagrams and hypergeometric functions is discussed. Special attention is devoted to existing techniques for the constructionof the Y-expansion. As an example, we present a detailed discussion of the construction of the Y-expansion of the Appell function 3 around rational values of parameters via an iterative solution of differential equations. Another interesting example is the Puiseux-type solution involving a differential operator generated by a hypergeometric… 
Olsson.wl : a Mathematica package for the computation of linear transformations of multivariable hypergeometric functions
The Olsson.wl Mathematica package, which aims to find linear transformations for some classes of multivariable hypergeometric functions, is presented and a companion package, called ROC2.wl, dedicated to the derivation of the regions of convergence of doublehypergeometric series is provided.
Multiple Series Representations of N-fold Mellin-Barnes Integrals.
The first evaluation of the hexagon and double box conformal Feynman integrals with unit propagator powers is presented, and the method allows the determination of a single "master series" for each series representation, which considerably simplifies convergence studies and/or numerical checks.
Analytic continuation of Lauricella's function F D (N) for large in modulo variables near hyperplanes {z j = z l }
We consider the Lauricella hypergeometric function , depending on variables , and obtain formulas for its analytic continuation into the vicinity of a singular set which is an intersection of the
Analytic continuation of Lauricella's function F D (N) for variables close to unit near hyperplanes {z j = z l }
For the Lauricella hypergeometric function with an arbitrary number of variables , we construct formulas for analytic continuation into the vicinity of hyperplanes and their intersections providing
Analytic Periods via Twisted Symmetric Squares
We study the symmetric square of Picard-Fuchs operators of genus one curves and the thereby induced generalized Clausen identities. This allows the computation of analytic expressions for the periods
14 72 1 v 2 [ he pth ] 2 5 O ct 2 02 1 Cohomology of Differential Forms and Feynman diagrams October 26 , 2021
  • 2021
Co-Homology of Differential Forms and Feynman Diagrams
In the present review we provide an extensive analysis of the intertwinement between Feynman integrals and cohomology theories in light of recent developments. Feynman integrals enter in several
Cohen-Macaulay Property of Feynman Integrals
: The connection between Feynman integrals and GKZ A -hypergeometric systems has been a topic of recent interest with advances in mathematical techniques and computational tools opening new
Hypergeometric Structures in Feynman Integrals
An automated method is devised which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions and solves these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series.


NumExp: Numerical epsilon expansion of hypergeometric functions
Algebraic A-hypergeometric functions and their monodromy
The study of hypergeometric functions started in 1813 with a paper by Gauss. Hypergeometric functions are generalizations of classical elementary functions such as arcsin and log. Around 1900,
Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams
The Gauss hypergeometric functions 2F1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown
Feynman integrals as A-hypergeometric functions
  • L. Cruz
  • Mathematics
    Journal of High Energy Physics
  • 2019
Abstract We show that the Lee-Pomeransky parametric representation of Feynman integrals can be understood as a solution of a certain Gel’fand-Kapranov-Zelevinsky (GKZ) system. In order to define
Feynman integrals and hyperlogarithms
We study Feynman integrals in the representation with Schwinger parameters and derive recursive integral formulas for massless 3- and 4-point functions. Properties of analytic (including dimensional)
Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction.
The algebraic and analytic structure of Feynman integrals is studied by proposing an operation that maps an integral into pairs of integrals obtained from a master integrand and a corresponding master contour, and it is demonstrated that it can be given a diagrammatic representation purely in terms of operations on graphs.
Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions
The goal of this paper is to raise the possibility that there exists a meaningful theory of `motives' associated to certain hypergeometric integrals, viewed as functions of their parameters. It goes
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It is well known that Feynman integrals in dimensional regularization often evaluate to functions of hypergeometric type. Inspired by a recent proposal for a coaction on one-loop Feynman integrals in