Finding new relationships between hypergeometric functions by evaluating Feynman integrals

TL;DR

This paper discovers new mathematical relationships between hypergeometric functions by analyzing Feynman integrals calculated through various methods, leading to novel reduction formulas and connections among special functions.

Contribution

It introduces new relationships and reduction formulas between hypergeometric functions derived from Feynman integral evaluations using different techniques.

Findings

New relation between Appell functions F_1 and F_4

Reduction of F_1 to _3F_2 hypergeometric function

Explicit formula linking F_1 and _2F_1 functions

Abstract

Several new relationships between hypergeometric functions are found by comparing results for Feynman integrals calculated using different methods. A new expression for the one-loop propagator-type integral with arbitrary masses and arbitrary powers of propagators is derived in terms of only one Appell hypergeometric function F_1. From the comparison of this expression with a previously known one, a new relation between the Appell functions F_1 and F_4 is found. By comparing this new expression for the case of equal masses with another known result, a new formula for reducing the F_1 function with particular arguments to the hypergeometric function _3F_2 is derived. By comparing results for a particular one-loop vertex integral obtained using different methods, a new relationship between F_1 functions corresponding to a quadratic transformation of the arguments is established. Another reduction formula for the F_1 function is found by analysing the imaginary part of the two-loop self-energy integral on the cut. An explicit formula relating the F_1 function and the Gaussian hypergeometric function _2F_1 whose argument is the ratio of polynomials of degree six is presented.