用离散时间热核计算某些三角character和的方法
On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel
摘要：
本文提出了一种通用的方法，它可以显式地求出在单位根$\mathbf{m}=(m_1,\ldots,m_d)$上求出的$d\geq 1$初等三角函数乘积的某些$n$次幂和。我们的方法是首先确定在考虑中的表达作为一个离散拉普拉斯算子的特征值相关联的图形，其顶点形成一个$d$ -维离散环面$G_{\mathbf{m}}$，这取决于$\mathbf{m}$。然后，所讨论的和与$G_{\mathbf{m}}$上的马尔可夫链的第$n$步相关。马尔可夫链允许解释为一个特殊的随机游走，也被看作是离散时间和离散空间的热扩散，因此所讨论的总和与相关热核的特殊值有关。我们的评价如下推导出的热核，这是通过周期化的无限格$\mathbb{Z}^{d}$覆盖$G_{\mathbf{m}}$的热核的组合表达式。
英文摘要：
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 $G_{\mathbf{m}}$.
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