Bernoulli数的新定义

New definition of Bernoulli numbers

无须利用幂级数而仅用定义或等幂和的一个递推公式直接给出Bernoulli数新的初等定义,其与经典的Bernoulli数相等。该定义的目的性明确且更加符合Jacobi Bernoulli原来的想法,即证明了 n-1(nk)Bk=0,B0=1 k=0(p+1Sp(n-1)=1p+1 pk)Bknp+1-k。

A primary and new definition of Bernoulli numbers has been given by a simple recursive formula for sums of power integers and without using the power series. The given Bernoulli numbers equal the classical Bernoulli numbers by proposed definition. Namely, n-1k=0(nk)B_k=0,B_0=1S_p(n-1)=1p+1pk=0((p+1)k)B_kn^(p+1-k) has been proved.