ar X iv : 0 90 4 . 16 01 v 1 [ m at h - ph ] 9 A pr 2 00 9 High order Fuchsian equations for the square lattice

We consider the Fuchsian linear differential equation obtained (modulo a prime) for χ̃(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 χ̃(1) and χ̃(3) can be removed from χ̃(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 LE , the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order terms χ̃(3) and χ̃(4). We conjecture that a linear differential operator equivalent to a symmetric (n− 1)-th power of LE occurs as a left-most factor in the minimal order linear differential operators for all χ̃(n)’s. PACS: 05.50.+q, 05.10.-a, 02.30.Hq, 02.30.Gp, 02.40.Xx AMS Classification scheme numbers: 34M55, 47E05, 81Qxx, 32G34, 34Lxx, 34Mxx, 14Kxx Key-words: Susceptibility of the Ising model, formal power series, long series expansions, Fuchsian linear differential equations, globally nilpotent linear differential operators, indicial equation, singular behavior, critical exponents, apparent singularities, formal modular calculations, diff-Padé series analysis, rational number reconstruction, elliptic functions, weight-1 modular forms. High order Fuchsian ODE for χ̃ 2