High order Fuchsian ODE for tilde \chi(5)

doi:10.1088/1751-8113/42/27/275209

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

A Bostan et al 2009 J. Phys. A: Math. Theor. 42 275209 (32pp) doi: 10.1088/1751-8113/42/27/275209

PDF (350 KB) | References

A Bostan¹, S Boukraa², A J Guttmann³, S Hassani⁴, I Jensen³, J-M Maillard⁵ and N Zenine⁴
¹ INRIA Paris-Rocquencourt, Domaine de Voluceau, B.P. 105 78153 Le Chesnay, Cedex, France
² LPTHIRM and Département d'Aéronautique, Université de Blida, Algeria
³ ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
⁴ Centre de Recherche Nucléaire d'Alger, 2 Bd. Frantz Fanon, BP 399, 16000 Alger, Algeria
⁵ LPTMC, UMR 7600 CNRS, Université de Paris, Tour 24, 4ème étage, case 121, 4 Place Jussieu, 75252 Paris Cedex 05, France
E-mail: alin.bostan@inria.fr, boukraa@mail.univ-blida.dz, tonyg@ms.unimelb.edu.au, I.Jensen@ms.unimelb.edu.au, maillard@lptmc.jussieu.fr, maillard@lptl.jussieu.fr and njzenine@yahoo.com

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.