The Fuchsian differential equation of the square lattice Ising model $χ(3)$ susceptibility

Abstract

Using an expansion method in the variables $ x_i$ that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the Ising square lattice model susceptibility $χ$, we generate a long series of coefficients for the 3-particle contribution $χ^{(3)}$, using a $ N^4$ polynomial time algorithm. We give the Fuchsian differential equation of order seven for $χ^{(3)}$ that reproduces all the terms of our long series. An analysis of the properties of this Fuchsian differential equation is performed.