High order Fuchsian equations for the square lattice Ising model: 5

We consider the Fuchsian linear differential equation obtained (modulo a prime) for , 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 and can be removed from 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 and . 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 's.