Lattice Green's functions in all dimensions

We give a systematic treatment of lattice Green functions (LGF) on the d-dimensional diamond, simple cubic, body-centred cubic and face-centred cubic lattices for arbitrary dimensionality d ≥ 2 for the first three lattices, and for 2 ≤ d ≤ 5 for the hyper-fcc lattice. We show that there is a close connection between the LGF of the d-dimensional hypercubic lattice and that of the $(d-1)$-dimensional diamond lattice. We give constant-term formulations of LGFs for all lattices and dimensions. Through a still under-developed connection with Mahler measures, we point out an unexpected connection between the coefficients of the s.c., b.c.c. and diamond LGFs and some Ramanujan-type formulae for 1/π.