On some mathematical identities resulting from evaluation of the partition function for an electron moving in a periodic lattice

TL;DR

This paper explores alternative mathematical methods for evaluating the partition function of an electron in a crystal lattice, revealing identities that connect physics with combinatorics and graph theory.

Contribution

It introduces new mathematical identities derived from physical models, bridging condensed matter physics with combinatorial and graph-theoretic analysis.

Findings

Derived new identities from partition function evaluations

Connected physical models with combinatorial mathematics

Provided alternative analytical approaches for lattice systems

Abstract

We consider a simple model of the dynamics of a single electron in a crystal lattice. Although this is a standard problem in condensed matter physics, alternative ways of evaluating a partition function for such a system lead to equalities, that may be interesting from the point of view of mathematical analysis, combinatorics and graph theory.