Hypergeometric Forms for Ising-Class Integrals

Abstract

We apply experimental-mathematical principles to analyze the integrals

C_{n,k} and:= \frac{1}{n!} \int_0^{\infty} \cdots \int_0^{\infty} \frac{dx_1 \, dx_2 \cdots \, dx_n}{(\cosh x_1 + \dots + \cosh x_n)^{k+1}.

These are generalizations of a previous integral $C_n := C_{n,1}$ relevant to the Ising theory of solid-state physics. We find representations of the $C_{n,k}$ in terms of Meijer $G$-functions and nested Barnes integrals. Our investigations began by computing 500-digit numerical values of $C_{n,k}$ for all integers $n, k$, where $n \in [2, 12]$ and $k \in [0,25]$. We found that some $C_{n,k}$ enjoy exact evaluations involving Dirichlet $L$-functions or the Riemann zeta function. In the process of analyzing hypergeometric representations, we found---experimentally and strikingly---that the $C_{n,k}$ almost certainly satisfy certain interindicial relations including discrete $k$-recurrences. Using generating functions, differential theory, complex analysis, and Wilf--Zeilberger algorithms we are able to prove some central cases of these relations.