Spanning tree generating functions and Mahler measures

TL;DR

This paper introduces spanning tree generating functions (STGFs) that unify spanning tree constants and lattice Green functions, expressing them as hypergeometric functions and Dirichlet L-series for various lattices, revealing new mathematical identities.

Contribution

It establishes a novel framework linking STGFs with Mahler measures, hypergeometric functions, and L-series, providing explicit formulas for spanning tree constants in multiple dimensions.

Findings

Closed-form hypergeometric expressions for all regular 2D and 3D lattice STGFs.

Representation of STGFs as Dirichlet L-series for various lattices.

Discovery of new integral identities and hypergeometric connections related to lattice Green functions.

Abstract

We define the notion of a spanning tree generating function (STGF) $\sum a_n z^n$, which gives the spanning tree constant when evaluated at $z=1,$ and gives the lattice Green function (LGF) when differentiated. By making use of known results for logarithmic Mahler measures of certain Laurent polynomials, and proving new results, we express the STGFs as hypergeometric functions for all regular two and three dimensional lattices (and one higher-dimensional lattice). This gives closed form expressions for the spanning tree constants for all such lattices, which were previously largely unknown in all but one three-dimensional case. We show for all lattices that these can also be represented as Dirichlet $L$-series. Making the connection between spanning tree generating functions and lattice Green functions produces integral identities and hypergeometric connections, some of which appear to be new.