## Lattice Green functions: the *d*-dimensional face-centred cubic lattice,

*d = 8, 9, 10, 11, 12*

Saoud Hassani, Christoph Koutschan, Jean-Marie Maillard, Nadjah Zenine

### Abstract

We previously reported on a recursive method to generate the expansion of the lattice Green function of the*d*-dimensional face-centred cubic lattice. The method was used to generate many coefficients for

*d = 7*and the corresponding linear differential equation has been obtained. We show the strength and the limit of the method by producing the series and the corresponding differential equations for

*d = 8, 9, 10, 11*. The differential Galois groups of these differential equations are shown to be symplectic for

*d = 8, 10*and orthogonal for

*d = 9, 11*.

### Paper

The full paper is available here.### Supplementary electronic material

Here we give the minimal-order differential operators for the lattice
Green functions of the face-centred cubic lattice, for different
dimensions *d*. The second column displays the order of the
operator, the third column the maximal degree of its coefficients, and the
fourth column shows the number of terms of the Taylor series that are
necessary to guess some non-minimal differential operators which allow to
obtain the minimal-order one as their greatest common right divisor. In
the sixth column, we give the recurrence equation for the
number *a(n)* of returning walks in the *d*-dimensional
f.c.c. lattice; the *n*-th Taylor coefficient of the lattice Green
function is equal to *a(n)/c ^{n}*, where

*c = 2d(d-1)*. Note that the operator for

*d = 12*is only available modulo the prime

*p = 2*.

^{31}- 1 d | order | degree | terms | operator | recurrence |

3 | 3 | 5 | 20 | fcc3_mop | fcc3_rec |

4 | 4 | 10 | 40 | fcc4_mop | fcc4_rec |

5 | 6 | 17 | 88 | fcc5_mop | fcc5_rec |

6 | 8 | 43 | 228 | fcc6_mop | fcc6_rec |

7 | 11 | 68 | 391 | fcc7_mop | fcc7_rec |

8 | 14 | 126 | 714 | fcc8_mop | fcc8_rec |

9 | 18 | 169 | 999 | fcc9_mop | fcc9_rec |

10 | 22 | 300 | 1739 | fcc10_mop | fcc10_rec |

11 | 27 | 409 | 2464 | fcc11_mop | |

12 | 32 | 617 | 3618 | fcc12_mop_mod |

In the following we provide the ODEs and PDEs (in operator notation) for *T*_{2}(*z,y*)
and *T*_{3}(*z,y*):

- ODE
*L*_{5}with respect to*z*for*T*_{2}(*z,y*): T2_ode_z.txt - ODE
*N*_{5}with respect to*y*for*T*_{2}(*z,y*): T2_ode_y.txt - The operator
*PDE*_{3}^{(1)}for*T*_{2}(*z,y*): T2_pde_3_1.txt - The operator
*PDE*_{3}^{(2)}for*T*_{2}(*z,y*): T2_pde_3_2.txt - The operator
*PDE*_{2}for*T*_{2}(*z,y*): T2_pde_2.txt - Gröbner basis of PDE for ann(
*T*_{2}): T2_pde_gb.txt - ODE
*L*_{9}with respect to*z*for*T*_{3}(*z,y*): T3_ode_z.txt - ODE
*N*_{9}with respect to*y*for*T*_{3}(*z,y*): T3_ode_y.txt - The operator
*PDE*_{4}^{(1)}for*T*_{3}(*z,y*): T3_pde_4_1.txt - The operator
*PDE*_{4}^{(2)}for*T*_{3}(*z,y*): T3_pde_4_2.txt - The operator
*PDE*_{3}for*T*_{3}(*z,y*): T3_pde_3.txt - Gröbner basis of PDE for ann(
*T*_{3}): T3_pde_gb.txt

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