Hypergeometric forms for Ising-class integrals

Borwein, Jonathan M. and Crandall, Richard E. and Bailey, David H. (2007) Hypergeometric forms for Ising-class integrals. Experimental Mathematics, 16 .

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    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.

    Item Type: Article
    Additional Information: pubdom TRUE
    Subjects: 33-xx Special functions
    65-xx Numerical analysis
    82-xx Statistical mechanics, structure of matter
    Faculty: UNSPECIFIED
    Depositing User: Users 1 not found.
    Date Deposited: 12 Jul 2006
    Last Modified: 11 Jan 2015 18:56

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