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05812430 Lin, Yang Zheng, Deyin Multiple convolution formulas on powers of positive integers. J. Hangzhou Norm. Univ., Nat. Sci. 8, No. 5, 327-330,349 (2009). 2009 Editorial Department of Hangzhou Normal University, Hangzhou, Zhejiang ZH multiple convolution generating function Stirling number of the second kind Summary: If a sum of $k$ positive integers equals $n$, 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 the 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.