MMN-3842

# Some identities for derangement numbers

**Sibel Koparal**, Bursa Uludağ University Department of Mathematics, 16059, Bursa, Turkey,`sibelkoparal1@gmail.com`

**Nese Ömür**, Kocaeli University Department of Mathematics, 41380, Kocaeli, Turkey,`neseomur@gmail.com`

**Kübra Nur Südemen**, Kocaeli University Department of Mathematics, 41380, Kocaeli, Turkey,`kubranursudemen@hotmail.com`

## Abstract

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

Vol. 23 (2022), No. 2, pp. 773-785

DOI: 10.18514/MMN.2022.3842