Experimental Mathematics

Hypergeometric Forms for Ising-Class Integrals

D. H. Bailey, D. Borwein, J. M. Borwein, and R. E. Crandall

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We apply experimental-mathematical principles to analyze the integrals

C_{n,k} and:= \frac{1}{n!} \int_0^{\infty} \cdots \int_0^{\infty} \frac{dx_1 \, dx_2 \cdots \, dx_n}{(\cosh x_1 + \dots + \cosh x_n)^{k+1}.

These are generalizations of a previous integral $C_n := C_{n,1}$ relevant to the Ising theory of solid-state physics. We find representations of the $C_{n,k}$ in terms of Meijer $G$-functions and nested Barnes integrals. Our investigations began by computing 500-digit numerical values of $C_{n,k}$ for all integers $n, k$, where $n \in [2, 12]$ and $k \in [0,25]$. We found that some $C_{n,k}$ 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 $C_{n,k}$ 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.

Article information

Experiment. Math., Volume 16, Issue 3 (2007), 257-276.

First available in Project Euclid: 7 March 2008

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65D30: Numerical integration

Numerical quadrature numerical integration arbitrary precision


Bailey, D. H.; Borwein, D.; Borwein, J. M.; Crandall, R. E. Hypergeometric Forms for Ising-Class Integrals. Experiment. Math. 16 (2007), no. 3, 257--276. https://projecteuclid.org/euclid.em/1204928528

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