The Hankel determinants for the generalized derangement polynomials of order r

This paper sets out to introduce the generalized derangement polynomials of order $r$. It then proceeds to establish various identities associated with these polynomials, along with providing recurrence relations for derangement polynomials of order $r$. Additionally, the paper offers a probabilistic approach for the generalized derangement polynomials of order $r$. Furthermore, it furnishes a clear expression for the Hankel determinants pertaining to these generalized derangement polynomials of order $r$, and subsequently infers the Hankel determinants for derangement polynomials of the same order, as well as for the count of cyclic derangements.