关于Lehmer级数注记

On Note of Lehmer Series

目的利用已知级数和级数的算数运算,尝试使用裂项的方法构造新的级数和级数恒等式。方法已知Lehmer级数恒等式,对其逐次裂项,得到含有1到5个因子的分式,然后将它们化成部分分式,通过一定程序将这些分式转化成分母含有1到5个因子的二项式系数倒数级数。结果使用裂项法构造出了一批新的分母含有1到5个奇因子的封闭性二项式系数的倒数级数,然后利用反三角函数与反双曲函数的关系得到了一批分母含有奇因子交错的二项式系数倒数级数恒等式和二项式系数倒数值级数恒等式。结论利用已知级数使用裂项的方法研究二项式系数变换是组合分析的新手段,也是产生新级数的一种方法。

Objective To structure new series and series identities by splitting items via the known series and series of arithmetic operation.Methods Given the known series identities——Lehmer series,fractions with 1 to 5 factors by successive splitting items are obtained.First,these fractions are put into partial fraction;then,these series of reciprocals of binomial coefficients have got 1 to 5 factors through certain procedures.Results Several new series of reciprocals of binominal coefficients can be structured by splitting items.These denominators of series contain different multiplications of 1 to 5 odd factors.The staggered series of reciprocals of binominal coefficients are presented by inverse trigonometric function and inverse hyperbolic function.And some series of number values of reciprocals of binominal coefficients are given.Conclusion Given the known series identities,the method of split items offers a new way of combinatorial analysis.Meanwhile it is a new method for structuring new series.