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.


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
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