Generalized Central Factorial Numbers with Odd Arguments

Youmna H Zaid,B S El-Desouky,F A Shiha

doi:10.4236/ojmsi.2020.83005

Abstract

In this paper, we consider r-generalization of the central factorial numbers with odd arguments of the first and second kind. Mainly, we obtain various identities and properties related to these numbers. Matrix representation and the relation between these numbers and Pascal matrix are given. Furthermore, the distributions of the signless r-central factorial numbers are derived. In addition, connections between these numbers and the Legendre-Stirling numbers are given.

Highlights

Riordan ([1], pp. 213-217), defined the central factorial numbers of the first and second kind t (n; k ) and T (n; k ), respectively ∏ ∑ x n −1 i =1 x + n 2 − i n = k =0 t ( n, k ) xk (1)
We consider a polynomial generalization of the cental factorial numbers with odd arguments of the first and second kind, which we will denote by vr (n, k ) and Vr (n, k ), respectively
We prove that the central factorial numbers with odd arguments can be expressed in terms of r-central factorial numbers with odd arguments and vice versa

Summary

Introduction

Riordan ([1], pp. 213-217), defined the central factorial numbers of the first and second kind t (n; k ) and T (n; k ) , respectively. Kim et al [3] extended T (n, k ) to the r-central factorial numbers of the second kind, r is a non-negative integer. In [4], the central factorial numbers with even arguments of both kinds are given by. The central factorial numbers with odd arguments of both kinds are given by v (n,= k ) 4n−k t (2n +1, 2k +1) and V (n,= k ) 4n−k T (2n +1, 2k +1). We consider a polynomial generalization of the cental factorial numbers with odd arguments of the first and second kind, which we will denote by vr (n, k ) and Vr (n, k ) , respectively. The distribution of the signless r-central factorial numbers with odd arguments of the first kind is derived. We give many properties of these new numbers, including a new and interesting connection between these numbers and the Legendre-Stirling numbers

The Generalized Central Factorial Numbers with Odd Arguments

Generating Function Formulas

The Generalized Central Factorial Matrices with Odd Arguments

The Generalized Central Factorial Numbers and Legender-Stirling Numbers

Conclusion