*(English)*Zbl 1113.82020

Summary: In a previous paper [J. Phys. A 37, No. 41, 9651–9668 (2004; Zbl 1073.82014)] we gave the Fuchsian linear differential equation satisfied by ${\chi}^{\left(3\right)}$, the ‘three-particle’ contribution to the susceptibility of the isotropic square lattice Ising model. This paper gives the details of the calculations (with some useful tricks and tools) which allow one to obtain a long series in polynomial time. The method is based on series expansion in the variables that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the isotropic square lattice Ising model susceptibility $\chi $. The integration rules are straightforward due to remarkable formulae we derive for these variables.

We obtain without any numerical approximation ${\chi}^{\left(3\right)}$ as a fully integrated series in the variable $w=s/2/(1+{s}^{2})$, where $s=sinh\left(2K\right)$, with $K=J/kT$ the conventional Ising model coupling constant. We also give some perspectives and comments on these results.