Translator Disclaimer
2007 Hypergeometric Forms for Ising-Class Integrals
D. H. Bailey, D. Borwein, J. M. Borwein, R. E. Crandall
Experiment. Math. 16(3): 257-276 (2007).

Abstract

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.

Citation

Download Citation

D. H. Bailey. D. Borwein. J. M. Borwein. R. E. Crandall. "Hypergeometric Forms for Ising-Class Integrals." Experiment. Math. 16 (3) 257 - 276, 2007.

Information

Published: 2007
First available in Project Euclid: 7 March 2008

zbMATH: 1134.33016
MathSciNet: MR2367317

Subjects:
Primary: 65D30

Rights: Copyright © 2007 A K Peters, Ltd.

JOURNAL ARTICLE
20 PAGES


SHARE
Vol.16 • No. 3 • 2007
Back to Top