Lattice Green functions in all dimensions

TL;DR

This paper provides a comprehensive analysis of lattice Green functions across various dimensions and lattice types, revealing new connections with Mahler measures and Ramanujan-type formulas for 1/π.

Contribution

It introduces systematic formulations of LGFs for multiple lattices and dimensions, and uncovers novel links to Mahler measures and Ramanujan-type formulas.

Findings

Established constant-term formulations for LGFs across lattices and dimensions.

Discovered a connection between LGF coefficients and Ramanujan-type formulas for 1/π.

Linked LGFs of hypercubic and diamond lattices across dimensions.

Abstract

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 $d \ge 2$ for the first three lattices, and for $2 \le d \le 5$ 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 $1/\pi.$