Abstract
Recently, extended r-central factorial numbers of the second kind and extended r-central Bell polynomials were introduced and various results of them were investigated. The purpose of this paper is to further derive properties, recurrence relations and identities related to these numbers and polynomials using umbral calculus techniques. Especially, we will represent the extended r-central Bell polynomials in terms of quite a few families of well-known special polynomials.
Highlights
Introduction and preliminariesIn [5], the extended r-central factorial numbers of the second kind and the extended rcentral Bell polynomials were introduced and various properties and identities related to these numbers and polynomials were investigated by means of generating functions
The extended r-central Bell polynomials are an extended version of the central Bell polynomials and a central analogue of r-Bell polynomials
The special polynomials are Bernoulli polynomials, Euler polynomials, falling factorial polynomials, Abel polynomials, ordered Bell polynomials, Laguerre polynomials, Daehee polynomials, Hermite polynomials, polynomials closely related to the reverse Bessel polynomials and studied by Carlitz, and Bernoulli polynomials of the second kind
Summary
K, with n ≥ k, the central factorial numbers of the second kind T(n, k) are given by the coefficients in the expansion (see [1, 4,5,6, 10, 12, 20]). The central factorial numbers of the second kind T(n, k) are given by the generating function t e2. T(n, k) were generalized to the extended r-central factorial numbers of the second kind. The extended r-central factorial numbers of the second kind, Tr(n + r, k + r), are given by the generating function. The central Bell polynomials are given by the following Dobinski-like formula (see [12]):. The extended r-central Bell polynomials are given by the following Dobinskilike formula:.
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