An Euler-Maclaurin formula for polygonal sums

块计数约束调和和：谱展开与块定向欧拉-麦克劳林公式

Block-count constrained harmonic sums: spectral expansion and block-directed Euler-Maclaurin

本文精确刻画了固定数字块出现次数下调和和的性质，通过语言分解与随机基数展开建立谱理论，并给出介于欧拉-麦克劳林与泰勒展开之间的块定向插值公式。

Local asymptotic Euler-Maclaurin expansion for Riemann sums over a semi-rational polyhedron

Consider the Riemann sum of a smooth compactly supported function h(x) on a polyhedron in R^d, sampled at the points of the lattice Z^d/t. We give an asymptotic expansion when t goes to infinity, writing each coefficient of this expansion as a sum indexed by the faces f of the polyhedron, where the f-term is the integral over f of a differential operator applied to the function h(x). In particular, if a Euclidean scalar product is chosen, we prove that the differential operator for the face f can be chosen (in a unique way) to involve only normal derivatives to f. Our formulas are valid for a …

欧拉-麦克劳林公式的一种替代方法：仅通过积分近似求和

An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only

本文提出了一种不依赖高阶导数的替代求和公式，通过积分线性组合近似求和，并给出了余项的明确界，展示了其在计算效率和内存使用上的优势。

Euler-MacLaurin formulas via differential operators

Recently there has been a renewed interest in asymptotic Euler-MacLaurin formulas, partly due to applications to spectral theory of differential operators. Using elementary means, we recover such formulas for compactly supported smooth functions f on intervals, polygons, and 3-dimensional polytopes, where the coefficients in the asymptotic expansion are sums of differential operators involving only derivatives of f in directions normal to the faces of the polytope. Our formulas apply to wedges of any dimension. This paper builds on, and is motivated by, works of Guillemin, Sternberg, and other…

广义Euler-Maclaurin公式与Signature

Generalized Euler-Maclaurin formula and Signatures

本文将Euler-Maclaurin公式推广到路径几乎必然可求长的随机过程上的Riemann-Stieltjes和与积分，通过引入flip和sawtooth signature等新变体，证明Bernoulli数的对应物是重复分部积分中递归最小化误差的积分常数。

Ramanujan Summation and the Exponential Generating Function $ \sum_{k=0}^{\infty}\frac{z^{k}}{k!}ζ^{\prime}(-k)$

In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from w…