Hypergeometric Forms for Ising-Class Integrals

Abstract

We apply experimental-mathematical principles to analyze the integrals

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These are generalizations of a previous integral C_n := C_n,1 relevant to the Ising theory of solid-state physics [Bailey et al. 06]. We find representations of the C_n,k in terms of Meijer Gfunctions and nested Barnes integrals. Our investigations began by computing 500-digit numerical values of C_n,k for all integers n, k, where n ∈ [2, 12] and k ∈ [0, 25]. We found that some C_n,k enjoy exact evaluations involving Dirichlet Lfunctions 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.

Keywords:

• Primary 65D30
• Numerical quadrature
• numerical integration
• arbitrary precision