Hypergeometric Forms for Ising-Class Integrals
Bailey, D. H. ; Borwein, D. ; Borwein, J. M. ; Crandall, R. E.
Experiment. Math., Tome 16 (2007) no. 1, p. 257-276 / Harvested from Project Euclid
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.
Publié le : 2007-05-15
Classification:  Numerical quadrature,  numerical integration,  arbitrary precision,  65D30
@article{1204928528,
     author = {Bailey, D. H. and Borwein, D. and Borwein, J. M. and Crandall, R. E.},
     title = {Hypergeometric Forms for Ising-Class Integrals},
     journal = {Experiment. Math.},
     volume = {16},
     number = {1},
     year = {2007},
     pages = { 257-276},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1204928528}
}
Bailey, D. H.; Borwein, D.; Borwein, J. M.; Crandall, R. E. Hypergeometric Forms for Ising-Class Integrals. Experiment. Math., Tome 16 (2007) no. 1, pp.  257-276. http://gdmltest.u-ga.fr/item/1204928528/