scholarly journals Some identities on r-central factorial numbers and r-central Bell polynomials

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Dojin Kim ◽  
Taekyun Kim
Keyword(s):  
Bell Polynomials ◽  
Central Factorial Numbers

scholarly journals On Central Complete and Incomplete Bell Polynomials I

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 288 ◽  
Author(s):  
Taekyun Kim ◽  
Dae Kim ◽  
Gwan-Woo Jang
Keyword(s):  
Bell Polynomials ◽  
Explicit Formulas ◽  
Central Factorial Numbers

In this paper, we introduce central complete and incomplete Bell polynomials which can be viewed as generalizations of central Bell polynomials and central factorial numbers of the second kind, and also as ’central’ analogues for complete and incomplete Bell polynomials. Further, some properties and identities for these polynomials are investigated. In particular, we provide explicit formulas for the central complete and incomplete Bell polynomials related to central factorial numbers of the second kind.

scholarly journals On r-Central Incomplete and Complete Bell Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 724 ◽  
Author(s):  
Dae San Kim ◽  
Han Young Kim ◽  
Dojin Kim ◽  
Taekyun Kim
Keyword(s):  
Stirling Numbers ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Central Factorial Numbers

Here we would like to introduce the extended r-central incomplete and complete Bell polynomials, as multivariate versions of the recently studied extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, and also as multivariate versions of the r- Stirling numbers of the second kind and the extended r-Bell polynomials. In this paper, we study several properties, some identities and various explicit formulas about these polynomials and their connections as well.

scholarly journals On degenerate central complete bell polynomials

2019 ◽  
Vol 13 (3) ◽  
pp. 805-818
Author(s):  
Taekyun Kim ◽  
San Kim ◽  
Gwan-Woo Jang
Keyword(s):  
Bell Polynomials ◽  
Explicit Formulas ◽  
Central Factorial Numbers

In this paper, we consider of generalized central complete and incomplete Bell polynomials called degenerate central complete and incomplete Bell polynomials. These polynomials are generalizations of the recently introduced central complete Bell polynomials and `degenerate' analogues for the central complete and incomplete Bell polynomials. We investigate some properties and identities for these polynomials. Especially, we give explicit formulas for the degenerate central complete and incomplete Bell polynomials related to degenerate central factorial numbers of the second kind.

scholarly journals Extended Degenerate r-Central Factorial Numbers of the Second Kind and Extended Degenerate r-Central Bell Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 595 ◽  
Author(s):  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Taekyun Kim ◽  
Dojin Kim
Keyword(s):  
Bell Polynomials ◽  
Central Factorial Numbers

In this paper, we introduce the extended degenerate r-central factorial numbers of the second kind and the extended degenerate r-central Bell polynomials. They are extended versions of the degenerate central factorial numbers of the second kind and the degenerate central Bell polynomials, and also degenerate versions of the extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, all of which have been studied by Kim and Kim. We study various properties and identities concerning those numbers and polynomials and also their connections.

scholarly journals Unified Degenerate Central Bell Polynomials

2019 ◽  
Author(s):  
Mehmet Acikgoz ◽  
Ugur Duran
Keyword(s):  
Explicit Formula ◽  
Bernstein Polynomials ◽  
Summation Formula ◽  
Bell Polynomials ◽  
Genocchi Numbers ◽  
Central Factorial Numbers ◽  
Degenerate Bernstein Polynomials

In this paper, we firstly consider extended degenerate central factorial numbers of the second kind and provide some properties of them. We then introduce unified degenerate central Bell polynomials and numbers and investigate many relations and formulas including summation formula, explicit formula and derivative property. Moreover, we derive several correlations for the fully degenerate central Bell polynomials associated with the degenerate Bernstein polynomials and the degenerate Bernoulli, Euler and Genocchi numbers.

scholarly journals Note on $ r $-central Lah numbers and $ r $-central Lah-Bell numbers

2022 ◽  
Vol 7 (2) ◽  
pp. 2929-2939
Author(s):  
Hye Kyung Kim ◽  
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Bell Numbers ◽  
Explicit Formulas ◽  
Riemann Integral ◽  
Enumerative Combinatorics ◽  
Central Factorial Numbers ◽  
Number Counts

<abstract><p>The $ r $-Lah numbers generalize the Lah numbers to the $ r $-Stirling numbers in the same sense. The Stirling numbers and the central factorial numbers are one of the important tools in enumerative combinatorics. The $ r $-Lah number counts the number of partitions of a set with $ n+r $ elements into $ k+r $ ordered blocks such that $ r $ distinguished elements have to be in distinct ordered blocks. In this paper, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers ($ r\in \mathbb{N} $) are introduced parallel to the $ r $-extended central factorial numbers of the second kind and $ r $-extended central Bell polynomials. In addition, some identities related to these numbers including the generating functions, explicit formulas, binomial convolutions are derived. Moreover, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers are shown to be represented by Riemann integral, respectively.</p></abstract>

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
Author(s):  
Eda Yuluklu ◽  
Yilmaz Simsek ◽  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Special Polynomials ◽  
Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

scholarly journals Explicit, Determinantal, and Recurrent Formulas of Generalized Eulerian Polynomials

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 37
Author(s):  
Yan Wang ◽  
Muhammet Cihat Dağli ◽  
Xi-Min Liu ◽  
Feng Qi
Keyword(s):  
General Formula ◽  
Bell Polynomials ◽  
Eulerian Polynomials ◽  
Differentiable Functions ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials.

scholarly journals Some identities for degenerate complete and incomplete r-Bell polynomials

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jongkyum Kwon ◽  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim
Keyword(s):  
Bell Polynomials
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