On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel

TL;DR

This paper introduces a method to explicitly evaluate certain sums of powers of products of elementary trigonometric functions at roots of unity by linking them to heat kernels on discrete tori and Markov chains.

Contribution

It develops a novel approach connecting trigonometric sums to heat kernel analysis on discrete tori via eigenvalues of a graph Laplacian.

Findings

Explicit evaluation formulas for trigonometric sums derived

Connection established between sums and heat kernel on discrete tori

Method applicable to sums involving multiple elementary functions

Abstract

In this article we develop a general method by which one can explicitly evaluate certain sums of $n$-th powers of products of $d\geq 1$ elementary trigonometric functions evaluated at $\mathbf{m}=(m_1,\ldots,m_d)$-th roots of unity. Our approach is to first identify the individual terms in the expression under consideration as eigenvalues of a discrete Laplace operator associated to a graph whose vertices form a $d$-dimensional discrete torus $G_{\mathbf{m}}$ which depends on $\mathbf{m}$. The sums in question are then related to the $n$-th step of a Markov chain on $G_{\mathbf{m}}$. The Markov chain admits the interpretation as a particular random walk, also viewed as a discrete time and discrete space heat diffusion, so then the sum in question is related to special values of the associated heat kernel. Our evaluation follows by deriving a combinatorial expression for the heat kernel, which is obtained by periodizing the heat kernel on the infinite lattice $\mathbb{Z}^{d}$ which covers $G_{\mathbf{m}}$.