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

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 ≥ 1 elementary trigonometric functions evaluated at m = (m 1 , . . . , 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 m which depends on m. The sums in question are then related to the n-th step of a Markov chain on G 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 Z d which covers G m .