Hypergeometric forms for Ising-class integrals

We apply experimental-mathematical principles to analyze the integrals [[unable to reproduce here]]. These are generalizations of a previous integral Cn := Cn,1 relevant to the Ising theory of solid-state physics [Bailey et al. 06]. 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 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.