High order Fuchsian equations for the square lattice Ising model: $\tildeχ^{(5)}$

Abstract

We consider the Fuchsian linear differential equation obtained (modulo a prime) for $\tildeχ^{(5)}$, 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 $\tildeχ^{(1)}$ and $\tildeχ^{(3)}$ can be removed from $\tildeχ^{(5)}$ 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 $\tildeχ^{(3)}$ and $\tildeχ^{(4)}$. 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 $\tildeχ^{(n)}$'s.