# High order Fuchsian equations for the square lattice Ising model: $\chi^{(6)}$

## TL;DR

This paper derives and factorizes a high-order Fuchsian differential equation for the six-particle contribution to the magnetic susceptibility of the square lattice Ising model, revealing its structure and relation to elliptic integrals.

## Contribution

It introduces a method to generate and factorize the differential equation for , showing its structure and connection to elliptic integrals, extending previous work on  and  contributions.

## Key findings

- The differential operator for  splits into smaller factors.
- The left-most factor is equivalent to the symmetric fifth power of an elliptic integral operator.
- The factors in the direct sum are reconstructed exactly from series data.

## Abstract

This paper deals with $\tilde{\chi}^{(6)}$, the six-particle contribution to the magnetic susceptibility of the square lattice Ising model. We have generated, modulo a prime, series coefficients for $\tilde{\chi}^{(6)}$. The length of the series is sufficient to produce the corresponding Fuchsian linear differential equation (modulo a prime). We obtain the Fuchsian linear differential equation that annihilates the "depleted" series $\Phi^{(6)}=\tilde{\chi}^{(6)} - {2 \over 3} \tilde{\chi}^{(4)} + {2 \over 45} \tilde{\chi}^{(2)}$. The factorization of the corresponding differential operator is performed using a method of factorization modulo a prime introduced in a previous paper. The "depleted" differential operator is shown to have a structure similar to the corresponding operator for $\tilde{\chi}^{(5)}$. It splits into factors of smaller orders, with the left-most factor of order six being equivalent to the symmetric fifth power of the linear differential operator corresponding to the elliptic integral $E$. The right-most factor has a direct sum structure, and using series calculated modulo several primes, all the factors in the direct sum have been reconstructed in exact arithmetics.