scholarly journals Hypergeometric series representations of Feynman integrals by GKZ hypergeometric systems

2020 ◽  
Vol 2020 (4) ◽  
Author(s):  
René Pascal Klausen
Keyword(s):  
Hypergeometric Series ◽  
Feynman Integrals ◽  
Series Representations

scholarly journals Matrix coefficients of the large discrete series representations of Sp(2; R) as hypergeometric series of two variables

2012 ◽  
Vol 208 ◽  
pp. 201-263
Author(s):  
Takayuki Oda
Keyword(s):  
Differential Equations ◽  
Hypergeometric Series ◽  
Discrete Series ◽  
Higher Order ◽  
Holonomic System ◽  
Radial Part ◽  
Matrix Coefficients ◽  
Series Representations ◽  
The Matrix ◽  
Discrete Series Representations

AbstractWe investigate the radial part of the matrix coefficients with minimal K-types of the large discrete series representations of Sp(2; R). They satisfy certain difference-differential equations derived from Schmid operators. This system is reduced to a holonomic system of rank 4, which is finally found to be equivalent to higher-order hypergeometric series in the sense of Appell and Kampé de Fériet.

scholarly journals Yangian bootstrap for massive Feynman integrals

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Florian Loebbert ◽  
Julian Miczajka ◽  
Dennis Müller ◽  
Hagen Münkler
Keyword(s):  
Momentum Space ◽  
Conformal Symmetry ◽  
Feynman Integrals ◽  
Series Representations ◽  
Yangian Symmetry

We extend the study of the recently discovered Yangian symmetry of massive Feynman integrals and its relation to massive momentum space conformal symmetry. After proving the symmetry statements in detail at one and two loop orders, we employ the conformal and Yangian constraints to bootstrap various one-loop examples of massive Feynman integrals. In particular, we explore the interplay between Yangian symmetry and hypergeometric expressions of the considered integrals. Based on these examples we conjecture single series representations for all dual conformal one-loop integrals in D spacetime dimensions with generic massive propagators.

scholarly journals Infinite families of accelerated series for some classical constants by the Markov-WZ Method

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Mohamud Mohammed
Keyword(s):  
Hypergeometric Series ◽  
Series Representations ◽  
International Audience ◽  
Wz Method

International audience In this article we show the Markov-WZ Method in action as it finds rapidly converging series representations for a given hypergeometric series. We demonstrate the method by finding new representations for log(2),ζ (2) and ζ (3).

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

scholarly journals Higher depth mock theta functions and q-hypergeometric series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joshua Males ◽  
Andreas Mono ◽  
Larry Rolen
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Mock Modular Forms ◽  
Harmonic Maass Forms

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

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