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 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>

On a generalization of the Rényi–Srivastava characterization of the Poisson law

2021 ◽  
Vol 58 (1) ◽  
pp. 68-82
Author(s):  
Jean-Renaud Pycke
Keyword(s):  
Life Insurance ◽  
Collective Model ◽  
New Method ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Compound Distributions ◽  
Poisson Law ◽  
Pierre Loti

AbstractWe give a new method of proof for a result of D. Pierre-Loti-Viaud and P. Boulongne which can be seen as a generalization of a characterization of Poisson law due to Rényi and Srivastava. We also provide explicit formulas, in terms of Bell polynomials, for the moments of the compound distributions occurring in the extended collective model in non-life insurance.

scholarly journals Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers

2021 ◽  
Vol 19 (1) ◽  
pp. 833-849
Author(s):  
Feng Qi ◽  
Muhammet Cihat Dağlı ◽  
Dongkyu Lim
Keyword(s):  
Bell Polynomials ◽  
Explicit Formulas ◽  
Cauchy Numbers ◽  
Faà Di Bruno Formula

Abstract In this paper, with the aid of the Faà di Bruno formula and by virtue of properties of the Bell polynomials of the second kind, the authors define a kind of notion of degenerate Narumi numbers and polynomials, establish explicit formulas for degenerate Narumi numbers and polynomials, and derive explicit formulas for the Narumi numbers and polynomials and for (degenerate) Cauchy numbers.

scholarly journals Some Identities for a Sequence of Unnamed Polynomials Connected with the Bell Polynomials

2017 ◽  
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Bai-Ni Guo
Keyword(s):  
Stirling Numbers ◽  
Higher Order ◽  
Recursive Relation ◽  
Order Derivative ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Derivative Formula ◽  
Higher Order Derivative ◽  
Determinantal Expression

In the paper, using two inversion theorems for the Stirling numbers and binomial coecients, employing properties of the Bell polynomials of the second kind, and utilizing a higher order derivative formula for the ratio of two dierentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, and recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.

Some properties of the Hermite polynomials

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

Abstract In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

scholarly journals Some Properties of a Sequence Similar to Generalized Euler Numbers

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Haiqing Wang ◽  
Guodong Liu
Keyword(s):  
Generating Function ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Explicit Formulas ◽  
Central Factorial Numbers

We introduce the sequence {Un(x)} given by generating function (1/(et+e-t-1))x=∑n=0∞Un(x)(tn/n!)  (|t|<(1/3)π,1x:=1) and establish some explicit formulas for the sequence {Un(x)}. Several identities involving the sequence {Un(x)}, Stirling numbers, Euler polynomials, and the central factorial numbers are also presented.

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