Euler–Boole summation

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

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

The concept is named after Leonhard Euler and George Boole.

The periodic Euler functions are

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

The Euler–Boole formula to sum alternating series is

$\sum _{j=a}^{n-1}(-1)^{j}f(j+h)={\frac {1}{2}}\sum _{k=0}^{m-1}{\frac {E_{k}(h)}{k!}}\left((-1)^{n-1}f^{(k)}(n)+(-1)^{a}f^{(k)}(a)\right)+{\frac {1}{2(m-1)!}}\int _{a}^{n}f^{(m)}(x){\widetilde {E}}_{m-1}(h-x)\,dx,$

where $a,m,n\in \mathbb {N} ,a<n,h\in [0,1]$ 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, JSTOR 40391116
• Nico M. Temme: Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, 2011, ISBN 9781118030813, pp. 17–18