论文标题
关于使用离散时间热内核评估某些三角特征总和的方法
On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel
论文作者
论文摘要
在本文中，我们开发了一种通用方法，通过该方法可以明确评估$ n $ th $ d \ geq 1 $基本三角函数的某些总和在$ \ mathbf {m_1，m_1，\ ldots，m_d，m_d，m_d）$ - Unity的Th Roots of Unity。我们的方法是首先将所考虑的表达式中的单个术语识别为与图形相关的离散拉普拉斯操作员的特征值，该图形形成$ d $ d $ d $ d $ d $ d $ d $ d $ d $ g _ {\ mathbf {m mathbf {m}} $，这取决于$ \ mathbf {m} $。然后，这些总和与$ g _ {\ mathbf {m}} $上的马尔可夫链的$ n $ THET步骤有关。马尔可夫链将解释视为特定的随机步行，也被视为离散时间和离散的空间热扩散，因此所讨论的总和与相关热内核的特殊值有关。我们的评估是通过得出热核的组合表达式来进行的，该表达式是通过对无限晶格上的热核置入无限晶格$ \ mathbb {z}^{d} $获得的，该核涵盖了$ g _ {\ mathbf {m {m}}} $。
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 $G_{\mathbf{m}}$.
