关于一种高阶算术几何级数的求和问题

On a Problem of Summing a Kind of Higher-Order Arithmetic-Geometric Progression

本文对形如的高阶算术几何级数研究了显式求和公式的构造问题，并给出了公式系列的速归生成法则。作为例子，对一类多项式系数的三角和计算提供了求和公式。

Let a be a real or complex number with and . Denote by S(p,j) the Stirlingnumber of the second kind,and by niop the j -th difference of S(p,j). this note presents a straight forward derivation of an explicit summation formula ofthe formwhere pis any given positive integer. Consequently,a pair of summation formulas are given ofthe trigonometric sums cos and sin,where f(x) is a polynomial in x ofdegree p with real confficients and f(O) = being real with .Also expounded isa recursive process for the construction of the sequence of formulas for 1, 2,3, ...