Euler–Boole summation

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

2 e x t e t + 1 = ∑ n = 0 ∞ E n ( x ) t n n ! . {\displaystyle {\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

E ~ n ( x + 1 ) = − E ~ n ( x )  and  E ~ n ( x ) = E n ( x )  for  0 < x < 1. {\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

∑ j = a n − 1 ( − 1 ) j f ( j + h ) = 1 2 ∑ k = 0 m − 1 E k ( h ) k ! ( ( − 1 ) n − 1 f ( k ) ( n ) + ( − 1 ) a f ( k ) ( a ) ) + 1 2 ( m − 1 ) ! ∫ a n f ( m ) ( x ) E ~ m − 1 ( h − x ) d x , {\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 a , m , n ∈ N , a < n , h ∈ [ 0 , 1 ] {\displaystyle a,m,n\in \mathbb {N} ,a<n,h\in [0,1]} and f ( k ) {\displaystyle 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


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