Exact evaluations of some Meijer G-functions and probability of all eigenvalues real for product of two Gaussian matrices

TL;DR

This paper proves a conjecture about the algebraic structure of Meijer G-functions related to the probability that all eigenvalues are real in the product of two Gaussian matrices, revealing new identities involving these functions.

Contribution

It provides a proof of a conjecture on Meijer G-functions' structure and uncovers new identities, advancing understanding of eigenvalue distributions in random matrix products.

Findings

Proof of Forrester's conjecture on Meijer G-functions

New identities involving Meijer G-functions

Enhanced understanding of eigenvalue probabilities in Gaussian matrix products

Abstract

We provide a proof to a recent conjecture by Forrester [2014 J. Phys. A: Math. Theor. 47, 065202] regarding the algebraic and arithmetic structure of Meijer G-functions which appear in the expression for probability of all eigenvalues real for product of two real Gaussian matrices. In the process we come across several interesting identities involving Meijer G-functions.