Hypergeometric forms for ising-class integrals

We apply experimental -mathematical principles to analyze the integrals [GRAPHICS] These are generalizations of a previous integral C-n := C-n,C-1 relevant to the Ising theory of solid-state physics [Bailey et al. 06]. We find representations of the C-n,C-k in terms of Meijer G-functions and nested Barnes integrals. Our investigations began by computing 500-digit numerical values Of C-n,C-k for all integers n, k, where n is an element of [2,12].and k is an element of [0, 25]. We found that some C-n,C-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,C-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.