含有高阶广义调和数的无穷级数恒等式

Infinite Series Identities Containing High-Order Generalized Harmonic Numbers

在本文中,定义一类m阶广义调和数利用组合分析中的Abel分部求和法,我们将推导出一些含有2阶广义调和数和3阶广义调和数的无穷级数与交错级数恒等式。进一步对参数a和b取特殊值,获得一些新的关于π,π2,π3,ζ(3),Catalan常数和ln2的无穷级数表达式。

In this paper, we define a class of generalized harmonic numbers of order m by Applying Abel’s method on summation by parts in combination analysis, we shall derive several infinite series and alternating series containing generalized harmonic numbers of order 2 and 3. Furthermore, by selecting special values for parameters a and b, several new infinite series ex-pressions are obtained for π, π2, π3, ζ(3), Catalan constant and ln2 as consequences.