The r-central factorial numbers with even indices

TL;DR

This paper introduces the $r$-central factorial numbers with even indices, exploring their properties, identities, and distributions, and analyzing related matrices and factorizations.

Contribution

It extends central factorial numbers to include an $r$-parameter, providing new properties, identities, and distributional results for these generalized numbers.

Findings

The $r$-central factorial numbers are strictly log-concave.

They follow a Poisson-binomial distribution.

The paper discusses the factorization of related matrices.

Abstract

In this paper, we introduce the $r$-central factorial numbers with even indices of the first and second kind, as extended versions of the central factorial numbers with even indices of both kinds. We obtain several fundamental properties and identities related to these numbers. We show that the unsigned $r$-central factorial numbers with even indices of the first kind are strictly log-concave and Poisson-binomially distributed . Finally, we consider the $r$-central factorial matrices and the factorization of it.