由Lehmer级数导出二项式系数倒数级数

Series of Reciprocals of Binomial Coefficients by Lehmer Series

目的利用已知级数,尝试使用裂项的方法构造出新的级数和级数恒等式。方法首先对Lehmer级数积分一次,然后对其使用裂项法,构造出一批新的分母含有奇偶性不定的因子(m+i)(i=1,2,3,4,5)与1到4个奇因子乘积的二项式系数倒数级数。结果由Lehmer级数导出二项式系数倒数级数,利用反三角函数与反双曲函数关系给出交错二项式系数倒数和二项式系数倒数数值级数恒等式。结论给出已知级数恒等式,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有效而又简洁的方法。

Objective Utilizing the known series,try to structure new series and series identities by splitting items.Methods First,the Lehmer series are integrated.Then use splitting items.These denominators have parity indefinite factors multiplied by one to four odd factors and binominal coefficients.Results The series of reciprocals of binomial coefficients is induced by Lehmer series.Staggered reciprocal binomial coefficient is given by using the relation of the inverse trigonometric function and inverse hyperbolic function.And some numerical series of reciprocals of binominal coefficients are given.Conclusion Given the known series identities,the method of split items offers a new combinatorial analysis way to study binomial coefficient transformation.Meanwhile it is a kind of effective and simple method for structured new series.