当前位置: X-MOL 学术arXiv.cs.NA › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
A Generalization of Classical Formulas in Numerical Integration and Series Convergence Acceleration
arXiv - CS - Numerical Analysis Pub Date : 2021-06-06 , DOI: arxiv-2106.07621
Ibrahim Alabdulmohsin

Summation formulas, such as the Euler-Maclaurin expansion or Gregory's quadrature, have found many applications in mathematics, ranging from accelerating series, to evaluating fractional sums and analyzing asymptotics, among others. We show that these summation formulas actually arise as particular instances of a single series expansion, including Euler's method for alternating series. This new summation formula gives rise to a family of polynomials, which contain both the Bernoulli and Gregory numbers in their coefficients. We prove some properties of those polynomials, such as recurrence identities and symmetries. Lastly, we present one case study, which illustrates one potential application of the new expansion for finite impulse response (FIR) filters.

中文翻译:

数值积分和级数收敛加速中经典公式的推广

求和公式,例如欧拉-麦克劳林展开式或格雷戈里求积,在数学中有许多应用,从加速级数到评估分数和和分析渐近性等等。我们表明,这些求和公式实际上是作为单个级数展开的特定实例出现的,包括欧拉的交替级数方法。这个新的求和公式产生了一系列多项式,其系数中同时包含伯努利数和格雷戈里数。我们证明了这些多项式的一些性质,例如递归恒等式和对称性。最后,我们介绍了一个案例研究,它说明了有限脉冲响应 (FIR) 滤波器的新扩展的一种潜在应用。
更新日期:2021-06-15
down
wechat
bug