Linear transformations of Srivastava's $H_C$ triple hypergeometric function

TL;DR

This paper investigates linear transformations of Srivastava's $H_C$ triple hypergeometric function, linking it to Feynman integrals and deriving new transformation identities using Mellin-Barnes techniques.

Contribution

It introduces a method to derive linear transformations of $H_C$ from those of $_2F_1$ and $F_1$, applying the conic hull method to evaluate transformed Mellin-Barnes integrals.

Findings

Derived new linear transformations of $H_C$

Connected hypergeometric transformations to Feynman integrals

Validated transformations numerically against Feynman parametrization

Abstract

We explore the large set of linear transformations of Srivastava's $H_C$ triple hypergeometric function. This function has been recently linked to the massive one-loop conformal scalar 3-point Feynman integral. We focus here on the class of linear transformations of $H_C$ that can be obtained from linear transformations of the Gauss $_2F_1$ hypergeometric function and, as $H_C$ is also a three variable generalization of the Appell $F_1$ double hypergeometric function, from the particular linear transformation of $F_1$ known as Carlson's identity and some of its generalizations. These transformations are applied at the level of the 3-fold Mellin-Barnes representation of $H_C$. This allows us to use the powerful conic hull method of Phys. Rev. Lett. 127 (2021) no.15, 151601 for the evaluation of the transformed Mellin-Barnes integrals, which leads to the desired results. The latter can then be checked numerically against the Feynman parametrization of the conformal 3-point integral. We also show how this approach can be used to derive many known (and less known) results involving Appell double hypergeometric functions.