Euler–Boole summation is a method for summing alternating series based on Euler's polynomials, which are defined by
\( {\displaystyle {\frac {2e^{xt}}{e^{t}+1}}=\sum _{n=0}E_{n}(x){\frac {t^{n}}{n!}}.} \)
The concept is named after Leonhard Euler and George Boole.
The periodic Euler functions are
\( {\displaystyle {\widetilde {E}}_{n}(x+1)=-{\widetilde {E}}_{n}(x){\text{ and }}{\widetilde {E}}_{n}(x)=E_{n}(x){\text{ for }}0<x<1.} \)
The Euler–Boole formula to sum alternating series is
\( {\displaystyle \sum _{j=a}^{n-1}(-1)^{j}f(j+h)={\frac {1}{2}}\sum _{k=0}^{m-1}{\frac {E_{k}(h)}{k!}}\left((-1)^{n-1}f^{(k)}(n)+(-1)^{a}f^{(k)}(a)\right)+{\frac {1}{2(m-1)!}}\int _{a}^{n}f^{(m)}(x){\widetilde {E}}_{m-1}(h-x)\,dx,} \)
where \( {\displaystyle a,m,n\in \mathbb {N} ,a<n,h\in [0,1]} \) 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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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