含有2的幂次的Euler和的研究

Studies on Euler sums with power of 2

利用生成函数及特殊函数的积分,建立含有2n的Euler和与交错Euler和的关系,并系统地得到一些含有2n的Euler和的值。结果表明:权2,3的含有2n的Euler和可以用zeta值表示;权4的含有2n的Euler和可以用Li4(1/2)、ln(2)及zeta值表示;权5的两个含有2n的Euler和S4,1(1/2)、S122,1(1/2)可以分别用Li5(1/2)、Li4(1/2)、ln(2)及zeta值表示。

In this paper, generating functions and integrals of special functions are used to establish a relation between the Euler sums with 2^n and the alternating Euler sums, and some special Euler sums with power of 2 are obtained systematically. The results show that the Euler sums with 2^n of weights 2, 3 can be expressed with zeta values. The Euler sums with 2^n of weight 4 can be expressed with Li4（1/2）, ln（2） and zeta values. The two Euler sums with 2^n of weight 5, S4.1 （1/2）and S1^2 2,1（1/2）can be expressed with Li5（1/2 ）, Li4（1/2）, In（2）and the zeta values.