Hypergeometric forms for Ising-class integrals

TL;DR: It is found that some Cn,k enjoy exact evaluations involving Dirichlet Lfunctions or the Riemann zeta function and almost certainly satisfy certain interindicial relations including discrete k-recurrences.

Abstract: We apply experimental-mathematical principles to analyze integrals Cn. These are generalizations of a previous integral Cn := Cn,1 relevant to the Ising theory of solid-state physics [8]. We find representations of the Cn,k in terms of Meijer G-functions and nested-Barnes integrals. Our investigations began by computing 500-digit numerical values of Cn,k for all integers n, k where n ∈ [2, 12] and k ∈ [0, 25]. We found that some Cn,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 Cn,k almost certainly satisfy certain inter-indicial relations including discrete k-recursions. Using generating functions, differential theory, complex analysis, and Wilf-Zeilberger algorithms we are able to prove some central cases of these relations.