The Revisited q−Derangement Polynomials and Incomplete q−Derangement Polynomials

In this paper, the $q-$derangement polynomials of the second type $\mathfrak{d}_{\psi,q}(\gamma)$ and their generating function are defined. Subsequently, the incomplete $q-$derangement polynomials $d_{\psi,q}\left( \gamma,m\right)$ and numbers $d_{\psi,q}\left( m\right)$ are introduced, and various properties are derived. Furthermore, the relationship between the incomplete $q-$derangement numbers and $q-$ derangement numbers is demonstrated.