Abstract: In this paper we consider the integral-typefunctional downward of single death processes in the finite state space, including the Laplace transformation of its distribution and its polynomial moments as well as the distribution of staying times. As applications, a new proof for the recursive and explicit representation of high order moments of the first hitting times in the denumerable state space is presented; meanwhile, the estimates on the lower bound and the upper one of convergence rate in strong ergodicity are obtained.