Euler–Boole summation

Euler–Boole summation is a summation method for alternating series based on Euler's polynomials, which are defined by

{\frac {2e^{xt}}{e^{t}+1}}=\sum _{n=0}E_{n}(x){\frac {t^{n}}{n!}}.

The concept is named after Leonhard Euler and George Boole.

The periodic Euler functions are

{\tilde {E}}_{n}(x+1)=-{\tilde {E}}_{n}(x){\text{ and }}{\tilde {E}}_{n}(x)=E_{n}(x){\text{ for }}0<x<1.

The Euler–Boole formula to sum alternating series is

{\begin{aligned}&\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}{(m-1)!}}\int _{a}^{n}f^{(m)}(x){\tilde {E}}_{m-1}(h-x)\,dx,\end{aligned}}

where $a,m,n\in \mathbb {N}$ , $a<n$ , $h\in [0,1]$ and $f^{(k)}$ 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 (online,JSTOR)
Nico M. Temme: Special Functions: An Introduction to the Classical Functions of Mathematical Physics. Wiley, 2011, ISBN 9781118030813, pp. 17–18

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