Euler–Boole summation

Short description: Summation method for some divergent series

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

[math]\displaystyle{ \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]\displaystyle{ \widetilde E_n(x+1)=-\widetilde E_n(x)\text{ and } \widetilde E_n(x)=E_n(x) \text{ for } 0\lt x\lt 1. }[/math]

The Euler–Boole formula to sum alternating series is

[math]\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, }[/math]

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