High order Fuchsian equations for the square lattice Ising model: (χ)over-tilde(5)

We consider the Fuchsian linear differential equation obtained (modulo a prime) for (chi) over tilde ((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 (chi) over bar ((1)) and (chi) over tilde ((3)) can be removed from (chi) over tilde ((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 (chi) over tilde ((3)) and (chi) over tilde ((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. (chi) over tilde ((n))'s.