### Abstract

We obtain in exact arithmetic the order 24 linear differential operator L_{24} and the right-hand side E^{(5)} of the inhomogeneous equation L_{24}(Φ^{(5)}) = E^{(5)}, where φ^{(5)} = χ̃^{(5)} - χ̃^{(3)}/2 + χ̃^{(1)}/120 is a linear combination of n-particle contributions to the susceptibility of the square lattice Ising model. In Bostan et al (2009 J. Phys. A: Math. Theor. 42 275209), the operator L_{24} (modulo a prime) was shown to factorize into L^{(left)}
_{12} · L^{(right)}
_{12}; here we prove that no further factorization of the order 12 operator L^{(left)}
_{12} is possible. We use the exact ODE to obtain the behaviour of χ̃ ^{(5)} at the ferromagnetic critical point and to obtain a limited number of analytic continuations of χ̃^{(5)} beyond the principal disc defined by its high temperature series. Contrary to a speculation in Boukraa et al (2008 J. Phys. A: Math. Theor. 41 455202), we find that χ̃^{(5)} is singular at w = 1/2 on an infinite number of branches.

Original language | English |
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Article number | 195205 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 43 |

Issue number | 19 |

DOIs | |

Publication status | Published - 21 Apr 2010 |

Externally published | Yes |

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## Cite this

^{(5)}ODE in exact arithmetic.

*Journal of Physics A: Mathematical and Theoretical*,

*43*(19), [195205]. https://doi.org/10.1088/1751-8113/43/19/195205, https://doi.org/10.1088/1751-8113/43/19/195205