关于常义Dirichlet级数及Euler数的恒等式

The Identity on the Dirichlet Series and Euler Namber

本文得出一类常义 Dirichlet 级数及 Euler 数关于一般自然数的恒等式。

Abstract.Theorem.Suppose the function f(2)=SecZ= (E_(2n)/((2n)!)Z^(2n),where E_(2a) is a Ealer number,The Dirichlet Series D(2n+1): D(2n+1)=sum from 1=0 to ∞ ((-1)~1)/((21+1)^(2n+1))(n=0,1,2,…) (1) then for the natual namber n≥2m+1,we hare the identity: i)sum from k=0 to m b_(m,k)·(2k)!〔=n (E_(2n_1))/(2n_1)(E2n_2)/(2n_2)…(E_(2n_(2k+1)))/(2n_(2k+1)!)〕 (2) E_(2n+2m)/(2n)! ii)sum from k=1 to m b_(m,k) ((2k)!2^(4k))/(π^(2k))〔D(2n_1+1)D(2n_2+1)…D(2n_(2k+1)+1)〕 =((2n+2m)!2^(2m))/((2n)!π^(2m))D(2n+2m+1) (3) where b_(m,k)=(-1)^(m-k)(2k_1+1)~2(2k_2+1)~2…(2k_(m-k)+1)~2 (k=0,1,…,m-1),b_(m,m)=1.(4)