New Results on Higher-Order Changhee Numbers and Polynomials

Mathematical and Computer Modelling, 2011

In this paper new explicit expressions for both kinds of Comtet numbers and some interesting special cases are derived. Moreover, we define and study the generalized multiparameter non-central Stirling numbers and generalized Comtet numbers via differential operators. Furthermore, recurrence relations and new explicit formulas for those numbers are obtained. Finally some interesting special cases, new combinatorial identities and a connection between these numbers and some interesting polynomials are deduced.

International Journal of Mathematical Analysis, 2015

In this paper, the authors generate the Stirling numbers of third kind to find formula for the sum of several types of product of polynomials and polynomial factorials using inverse of the generalized difference operator of n th kind in the field of numerical analysis. Suitable examples are provided to illustrate the main results.

Bulletin of the Korean Mathematical Society, 2015

The Changhee polynomials and numbers are introduced in [6]. Some interesting identities and properties of those polynomials are derived from umbral calculus (see [6]). In this paper, we consider Witttype formula for the n-th twisted Changhee numbers and polynomials and derive some new interesting identities and properties of those polynomials and numbers from the Witt-type formula which are related to special polynomials.

arXiv: Number Theory, 2017

Recently, the numbers $Y_{n}(\lambda )$ and the polynomials $Y_{n}(x,\lambda)$ have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating functions. By using these generating functions with their functional equations and derivative equations, we derive various identities and relations including two recurrence relations, Vandermonde type convolution formula, combinatorial sums, the Bernstein basis functions, and also some well known families of special numbers and their interpolation functions such as the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of the first kind, and the zeta type function. Finally, by using Stirling's approximation for factorials, we investigate some approximation values of the special case of the numbers $Y_{n}\left( \lambda \right) $.

2020

1College of Mathematics and Physics, Inner Mongolia University for Nationalities, Tongliao 028043, Inner Mongolia, China 2Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, Henan, China 3School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China 4Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Largo San Leonardo Murialdo, 1, 00146-Roma, Italy 5Sezione di Matematica, International Telematic University UniNettuno, Corso Vittorio Emanuele II, 39, 00186-Roma, Italy

Journal of Integer Sequences, 2012

The generalized Stirling numbers S s;h (n, k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n, k; α, β, r) considered by Hsu and Shiue. From this relation, several properties of S s;h (n, k) and the associated Bell numbers B s;h (n) and Bell polynomials B s;h|n (x) are derived. The particular case s = 2 and h = −1 corresponding to the meromorphic Weyl algebra is treated explicitly and its connection to Bessel numbers and Bessel polynomials is shown. The dual case s = −1 and h = 1 is connected to Hermite polynomials. For the general case, a close connection to the Touchard polynomials of higher order recently introduced by Dattoli et al. is established, and Touchard polynomials of negative order are introduced and studied. Finally, a q-analogue S s;h (n, k|q) is introduced and first properties are established, e.g., the recursion relation and an explicit expression. It is shown that the q-deformed numbers S s;h (n, k|q) are special cases of the type-II p, qanalogue of generalized Stirling numbers introduced by Remmel and Wachs, providing the analogue to the undeformed case (q = 1). Furthermore, several special cases are discussed explicitly, in particular, the case s = 2 and h = −1 corresponding to the q-meromorphic Weyl algebra considered by Diaz and Pariguan.

Symmetry

The aim of this study was to define a new operator. This operator unify and modify many known operators, some of which were introduced by the author. Many properties of this operator are given. Using this operator, two new classes of special polynomials and numbers are defined. Many identities and relationships are derived, including these new numbers and polynomials, combinatorial sums, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, and the Changhee numbers. By applying the derivative operator to these new polynomials, derivative formulas are found. Integral representations, including the Volkenborn integral, the fermionic p-adic integral, and the Riemann integral, are given for these new polynomials.

Axioms, 2024

Recently, Appell-type polynomials have been investigated and applied in several ways. In this paper, we consider a new extension of Appell-type Changhee polynomials. We introduce two-variable generalized Appell-type λ-Changhee polynomials (2VGATλCHP). The generating function, series representations, and summation identities related to these polynomials are explored. Further, certain symmetry identities involving two-variable generalized Appell-type λ-Changhee polynomials are established. Finally, Mathematica was used to examine the zero distributions of two-variable truncated-exponential Appell-type Changhee polynomials.

2000

The object of this article is to present a generalization of stirling numbers and polynomials which were studied in a number of earlier work on the subject due to their importance for possible applications in certain problems arising in science and engineering (like curve fitting, coding theory, signal processing etc.). We prove that are result concerned the generalized stirling numbers

arXiv: Combinatorics, 2018

In the paper, the authors establish explicit formulas for the Dowling numbers and their generalizations in terms of generalizations of the Lah numbers and the Stirling numbers of the second kind. These results gen- eralize the Qi formula for the Bell numbers in terms of the Lah numbers and the Stirling numbers of the second kind.

Mathematical and Computational Applications

By using the generating functions for the generalized Stirling type numbers, Eulerian type polynomials and numbers of higher order, we derive various functional equations and differential equations. By using these equation, we derive some relations and identities related to these numbers and polynomials. Furthermore, by applying padic Volkenborn integral to these polynomials, we also derive some new identities for the generalized -Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials.

Journal of Inequalities and Applications

The aim of this is to give generating functions for new families of special numbers and polynomials of higher order. By using these generating functions and their functional equations, we derive identities and relations for these numbers and polynomials. Relations between these new families of special numbers and polynomials and Bernoulli numbers and polynomials are given. Finally, recurrence relations and derivative formulas, which are related to these numbers and polynomials, are given.

Journal Human Research in Rehabilitation, 2013

In this paper we give a representation of Stirling numbers of the second kind, we obtain explicit formulas for some cases of Stirling numbers of the second kind and illustrate a method for founding other such formulas. 2010 Mathematics Subject Classification. 11B73, 05A10.

2011

We define a generalization of the Stirling numbers of the second kind, which depends on two parameters. The matrices of integers that result are exponential Riordan arrays. We explore links to orthogonal polynomials by studying the production matrices of these Riordan arrays. Generalized Bell numbers are also defined, again depending on two parameters, and we determine the Hankel transform of these numbers.

2011

The first aim of this paper is to construct new generating functions for the generalized λ-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers, attached to Dirichlet character. We derive various functional equations and differential equations using these generating functions. The second aim is provide a novel approach to deriving identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations. Furthermore, by applying p-adic Volkenborn integral and Laplace transform, we derive some new identities for the generalized λ-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials. We also give many applications related to the class of these polynomials and numbers.