Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Hankel determinants for the generalized derangement polynomials of order r

Ghania Guettai, Diffalah Laissaoui, Mourad Rahmani

Abstract

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.

The Hankel determinants for the generalized derangement polynomials of order r

Abstract

This paper sets out to introduce the generalized derangement polynomials of order . It then proceeds to establish various identities associated with these polynomials, along with providing recurrence relations for derangement polynomials of order . Additionally, the paper offers a probabilistic approach for the generalized derangement polynomials of order . Furthermore, it furnishes a clear expression for the Hankel determinants pertaining to these generalized derangement polynomials of order , and subsequently infers the Hankel determinants for derangement polynomials of the same order, as well as for the count of cyclic derangements.
Paper Structure (4 sections, 16 theorems, 56 equations)

This paper contains 4 sections, 16 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. The generalized derangement polynomials of order $r$
  3. Probabilistic approach
  4. On the Hankel determinant of $r-$derangement polynomials

Key Result

Theorem 2.2

For $n\geq0$, we have the explicit formula of the generalized derangement polynomials of order $r$ where $r^{\overline{k}}$ is the rising factorial, defined by $r^{\overline{k}}=r(r+1)...(r+k-1)$, with $r^{\overline{0}}=1$.

Theorems & Definitions (28)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • ...and 18 more