@article{BBGHJMZ09,
author = {A. Bostan and S. Boukraa and A. J. Guttmann and S. Hassani and I. Jensen and J.-M. Maillard and N. Zenine},
title = {High order {F}uchsian equations for the square lattice {I}sing model: $\tilde{\chi}^{(5)}$},
journal = {J. Phys. A: Math. Theor.},
number = {27},
year = {2009},
volume = {42},
pages = {32pp},
doi = {10.1088/1751-8113/42/27/275209},
abstract = {We consider the Fuchsian linear differential equation obtained (modulo a prime) for $\tilde{\chi}^{(5)}$, the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of $\tilde{\chi}^{(1)}$ and $\tilde{\chi}^{(3)}$ can be removed from $\tilde{\chi}^{(5)}$ and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth order linear differential operator occurs as the left-most factor of the ``depleted" differential operator and it is shown to be equivalent to the symmetric fourth power of $L_E$, the linear differential operator corresponding to the elliptic integral $E$. This result generalizes what we have found for the lower order terms $\tilde{\chi}^{(3)}$ and $\tilde{\chi}^{(4)}$. We conjecture that a linear differential operator equivalent to a symmetric $(n-1)$-th power of $L_E$ occurs as a left-most factor in the minimal order linear differential operators for all $\tilde{\chi}^{(n)}$'s.},
}