Square lattice Ising model susceptibility: Series expansion method and differential equation for ${\chi }^{(3)}$

In a previous paper (J. Phys. A {\bf 37} (2004) 9651-9668) we have given the Fuchsian linear differential equation satisfied by ${\chi }^{(3)}$, 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) allowing one to obtain 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 formulas we derived for these variables. We obtain without any numerical approximation ${\chi }^{(3)}$ as a fully integrated series in the variable $w=s\mathrm{/}2\mathrm{/}(1+s^2)$, where $ s =sh (2K)$,with$K=J/kT$ the conventional Ising model coupling constant. We also give some perspectives and comments on these results.