Hypergeometric forms for Ising-class integrals.(English)Zbl 1134.33016

The Ising-class integrals under examination are defined by $C_{n,k}=\frac{1}{n!}\int_0^\infty \cdots \int_0^\infty\frac{dx_1\,dx_2\cdots dx_n} {(\cosh x_1+\cdots +\cosh x_n)^{k+1}}$ for positive integer $$n$$ and Re($$k)>-1$$. They can be expressed alternatively as integrals involving the Bessel function $$K_0$$ as $C_{n,k}=\frac{2^n}{n! k!}\int_0^\infty t^k\{K_0(t)\}^ndt.$ Explicit representations are known for $$n=1, 2$$ and in terms of the Meijer $$G$$ function when $$n=3, 4$$. No general results are available for $$n\geq 5$$. It is shown that for odd $$k\geq 1$$ there exist relations of the form $C_{3,k}=p_{3,k}+q_{3,k} L_{-3}(2), \qquad C_{4,k}=p_{4,k}+q_{4,k} \zeta(3),$ where the $$p$$ and $$q$$ coefficients are rational, $$L$$ is the Dirichlet $$L$$-sum $L_{-3}(2)=\sum_{m\geq 0}\{(3m+1)^{-2}-(3m+2)^{-2}\}$ and $$\zeta$$ is the Riemann zeta function. No such relations appear to exist for even values of $$k$$.
Based on numerical experimentation, it is conjectured that, for positive integer $$n$$ and $$M:=\lfloor(n+1)/2\rfloor$$, the integrals $$C_{n,k}$$ satisfy an order-M recurrence involving $$M+1$$ terms in the form $P_{n,0}(k)C_{n,k}+P_{n,1}(k)C_{n,k+2}+\cdots +P_{n,M}(k)C_{n,k+2M}=0,$ where $$P$$ are integral polynomials. Using generating functions, differential theory, complex analysis and Wilf-Zeilberger algorithms, the authors are able to prove some central cases of these relations.

MSC:

 33E20 Other functions defined by series and integrals 33C05 Classical hypergeometric functions, $${}_2F_1$$ 33F05 Numerical approximation and evaluation of special functions 65D30 Numerical integration
Full Text: