Английская Википедия:Euler–Boole summation
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Шаблон:Use American English Шаблон:Short description Euler–Boole summation is a method for summing alternating series based on Euler's polynomials, which are defined by
- <math>\frac{2e^{xt}}{e^t+1}=\sum_{n=0}^\infty E_n(x)\frac{t^n}{n!}.</math>
The concept is named after Leonhard Euler and George Boole.
The periodic Euler functions are
- <math>\widetilde E_n(x+1)=-\widetilde E_n(x)\text{ and } \widetilde E_n(x)=E_n(x) \text{ for } 0<x<1.</math>
The Euler–Boole formula to sum alternating series is
- <math>\sum_{j=a}^{n-1}(-1)^j f(j+h) = \frac{1}{2}\sum_{k=0}^{m-1} \frac{E_k(h)}{k!} \left((-1)^{n-1} f^{(k)}(n)+(-1)^a f^{(k)}(a)\right) + \frac 1 {2(m-1)!} \int_a^n f^{(m)}(x)\widetilde E_{m-1}(h-x) \, dx,</math>
where <math>a,m,n\in\N, a<n, h\in [0,1]</math> and <math>f^{(k)}</math> is the kth derivative.
References
- Jonathan M. Borwein, Neil J. Calkin, Dante Manna: "Euler–Boole Summation Revisited", The American Mathematical Monthly, Vol. 116, No. 5 (May, 2009), pp. 387–412, Шаблон:Jstor
- Nico M. Temme: Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, 2011, Шаблон:ISBN, pp. 17–18
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