Some identities for derangement numbers

Abstract

In this paper, we give some identities involving derangement numbers. In addition, we derive nonlinear differential equation from the generating function of r-derangement numbers and obtain some identities including generalized harmonic numbers and derangement numbers by using this differential equation. For example, for any positive integers N, n and r,Sigma(N)(k=0)Sigma(N-K)(i=0) (-1) ((r)(k)) ((r+N - i - k)(N - i -k) )d((r+N-i-k+1)n-r+k)/i!(n-r+k)! = D-r(n+N)/n!N!.