关于正整数幂的多重卷积公式

Multiple Convolution Formulas on Powers of Positive Integers

若k个正整数的和为n,那么这k个正整数积的r次幂的多重和就是正整数的r次幂的k重卷积.使用生成函数方法首先得到了一次幂和二次幂的k重卷积的求和公式,然后借助于导数算子和第二类Stirling数给出了一般的r次幂的k重卷积的求和公式.

If a sum of k positive integers equals to n, the multiple sum of the r-th powers of the product of these k positive integers is a k-fold convolution of the r-th powers of the positive integers. By means of generating function method, summation formulas of the k fold convolution of simple and double power are derived. Furthermore, the k-fold convolution formulas on the general r-th powers of positive integers are established with the derivative operator and the Stirling numbers of the second kind.