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Hypergeometric Forms for Ising-Class Integrals

  • Author(s): Bailey, David H.
  • Borwein, David
  • Borwein, Jonathan M.
  • Crandall, Richard E.
  • et al.
Abstract

We apply experimental-mathematical principles to analyze certain integrals relevant to the Ising theory of solid-state physics. We find representations of the these integrals in terms of Meijer G-functions and nested-Barnes integrals. Our investigations began by computing 500-digit numerical values of Cn,k, namely a 2-D array of Ising integrals for all integers n, k where n is in [2,12] and k is in [0,25]. We found that some Cn,k enjoy exact evaluations involving Dirichlet L-functions or the Riemann zeta function. In theprocess 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.

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