论文标题
多元伯努利多项式中的欧拉 - 麦克劳林求和公式
Euler-MacLaurin summation formula on polytopes and expansions in multivariate Bernoulli polynomials
论文作者
论文摘要
我们在多面体上提供了多维加权Euler--Maclaurin求和公式，以及由于L. J. Mordell在Bernoulli多项式中的串联扩展而引起的结果的多维概括。这些结果是更通用系列扩展的后果。也就是说，如果$χ_{τ\ Mathcal {p}} $表示扩张的整数凸出的特征函数polytope $ \ Mathcal {p} $和$ q $是一个适当的函数多项式。这些多元多项式与LERCH ZETA函数有关。为了证明我们的结果，我们需要仔细研究$ \ wideHat {qχ_{τ\ Mathcal {p}}} $的渐近扩展，$qχ_{τ\ nathcal {p}} $的傅立叶变换。
We provide a multidimensional weighted Euler--MacLaurin summation formula on polytopes and a multidimensional generalization of a result due to L. J. Mordell on the series expansion in Bernoulli polynomials. These results are consequences of a more general series expansion; namely, if $χ_{τ\mathcal{P}}$ denotes the characteristic function of a dilated integer convex polytope $\mathcal{P}$ and $q$ is a function with suitable regularity, we prove that the periodization of $qχ_{τ\mathcal{P}}$ admits an expansion in terms of multivariate Bernoulli polynomials. These multivariate polynomials are related to the Lerch Zeta function. In order to prove our results we need to carefully study the asymptotic expansion of $\widehat{qχ_{τ\mathcal{P}}}$, the Fourier transform of $qχ_{τ\mathcal{P}}$.
