scholarly journals Degenerate poly-Bell polynomials and numbers

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Hye Kyung Kim
Keyword(s):  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Genocchi Polynomials ◽  
Special Polynomials ◽  
Explicit Representations

AbstractNumerous mathematicians have studied ‘poly’ as one of the generalizations to special polynomials, such as Bernoulli, Euler, Cauchy, and Genocchi polynomials. In relation to this, in this paper, we introduce the degenerate poly-Bell polynomials emanating from the degenerate polyexponential functions which are called the poly-Bell polynomials when $\lambda \rightarrow 0$ λ → 0 . Specifically, we demonstrate that they are reduced to the degenerate Bell polynomials if $k = 1$ k = 1 . We also provide explicit representations and combinatorial identities for these polynomials, including Dobinski-like formulas, recurrence relationships, etc.

scholarly journals Fully degenerate Bell polynomials associated with degenerate Poisson random variables

2021 ◽  
Vol 19 (1) ◽  
pp. 284-296
Author(s):  
Hye Kyung Kim
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Poisson Random Variable ◽  
Central Moments ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

scholarly journals On Degenerate Truncated Special Polynomials

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 144 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Bernstein Polynomials ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Euler Polynomials ◽  
Exponential Polynomials ◽  
Special Polynomials ◽  
Summation Formulas ◽  
Special Cases ◽  
Proof Techniques

The main aim of this paper is to introduce the degenerate truncated forms of multifarious special polynomials and numbers and is to investigate their various properties and relationships by using the series manipulation method and diverse special proof techniques. The degenerate truncated exponential polynomials are first considered and their several properties are given. Then the degenerate truncated Stirling polynomials of the second kind are defined and their elementary properties and relations are proved. Also, the degenerate truncated forms of the bivariate Fubini and Bell polynomials and numbers are introduced and various relations and formulas for these polynomials and numbers, which cover several summation formulas, addition identities, recurrence relationships, derivative property and correlations with the degenerate truncated Stirling polynomials of the second kind, are acquired. Thereafter, the truncated degenerate Bernoulli and Euler polynomials are considered and multifarious correlations and formulas including summation formulas, derivation rules and correlations with the degenerate truncated Stirling numbers of the second are derived. In addition, regarding applications, by introducing the degenerate truncated forms of the classical Bernstein polynomials, we obtain diverse correlations and formulas. Some interesting surface plots of these polynomials in the special cases are provided.

scholarly journals On another approach for a family of Appell polynomials

Filomat ◽  
2018 ◽  
Vol 32 (12) ◽  
pp. 4155-4164
Author(s):  
H. Belbachir ◽  
Hadj Brahim ◽  
M. Rachidi
Keyword(s):  
Combinatorial Identities ◽  
Appell Polynomials ◽  
Genocchi Polynomials

The present study is devoted to some new formulas for a family of Appell polynomials. These formulas are expressed in terms of Bernoulli-Euler and Bernoulli-Genocchi polynomials. Moreover, other additive formulas and new combinatorial identities are established. In particular some closed relations with nested sums are exhibited.

scholarly journals Degenerate Catalan-Daehee numbers and polynomials of order $ r $ arising from degenerate umbral calculus

2021 ◽  
Vol 7 (3) ◽  
pp. 3845-3865
Author(s):  
Hye Kyung Kim ◽  
◽  
Dmitry V. Dolgy ◽  
Keyword(s):  
Bernoulli Polynomials ◽  
Umbral Calculus ◽  
Bell Polynomials ◽  
Psychological Burden ◽  
Special Polynomials ◽  
Sheffer Sequences ◽  
Special Numbers And Polynomials ◽  
Bernoulli Polynomials And Numbers ◽  
Euler Polynomials And Numbers ◽  
Degenerate Version

<abstract><p>Many mathematicians have studied degenerate versions of some special polynomials and numbers that can take into account the surrounding environment or a person's psychological burden in recent years, and they've discovered some interesting results. Furthermore, one of the most important approaches for finding the combinatorial identities for the degenerate version of special numbers and polynomials is the umbral calculus. The Catalan numbers and the Daehee numbers play important role in connecting relationship between special numbers.</p> <p>In this paper, we first define the degenerate Catalan-Daehee numbers and polynomials and aim to study the relation between well-known special polynomials and degenerate Catalan-Daehee polynomials of order $ r $ as one of the generalizations of the degenerate Catalan-Daehee polynomials by using the degenerate Sheffer sequences. Some of them include the degenerate and other special polynomials and numbers such as the degenerate falling factorials, the degenerate Bernoulli polynomials and numbers of order $ r $, the degenerate Euler polynomials and numbers of order $ r $, the degenerate Daehee polynomials of order $ r $, the degenerate Bell polynomials, and so on.</p></abstract>

scholarly journals Some Convolution Identities and an Inverse Relation Involving Partial Bell Polynomials

10.37236/2476 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Daniel Birmajer ◽  
Juan B. Gil ◽  
Michael D. Weiner
Keyword(s):  
Combinatorial Identities ◽  
Convolution Type ◽  
Binomial Coefficients ◽  
Bell Polynomials ◽  
Inverse Relation

We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting multinomial formula for the binomial coefficients. The inverse relation is deduced from a parametrization of suitable identities that facilitate dealing with nested compositions of partial Bell polynomials.

scholarly journals Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine functions and applications

2021 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Cosine Function ◽  
Binomial Coefficients ◽  
Series Expansions ◽  
Bell Polynomials ◽  
Specific Sequence ◽  
Hyperbolic Cosine ◽  
Faà Di Bruno Formula ◽  
Hyperbolic Cosine Function

Abstract In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine function and the inverse hyperbolic cosine function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

scholarly journals A new approach to Bell and poly-Bell numbers and polynomials

2021 ◽  
Vol 7 (3) ◽  
pp. 4004-4016
Author(s):  
Taekyun Kim ◽  
◽  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Hye Kyung Kim ◽  
...  
Keyword(s):  
Recurrence Relations ◽  
Bell Polynomials ◽  
Bell Numbers ◽  
New Approach ◽  
Special Polynomials ◽  
Explicit Expressions

<abstract><p>Bell polynomials are widely applied in many problems arising from physics and engineering. The aim of this paper is to introduce new types of special polynomials and numbers, namely Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. We also consider degenerate versions of those polynomials and numbers, namely degenerate Bell polynomials and numbers of the second kind and degenerate poly-Bell polynomials and numbers of the second kind, and deduce their similar results.</p></abstract>

scholarly journals Generalized Bell Polynomials and the Combinatorics of Poisson Central Moments

10.37236/541 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Nicolas Privault
Keyword(s):  
Poisson Distribution ◽  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Charlier Polynomials ◽  
Central Moments

We introduce a family of polynomials that generalizes the Bell polynomials, in connection with the combinatorics of the central moments of the Poisson distribution. We show that these polynomials are dual of the Charlier polynomials by the Stirling transform, and we study the resulting combinatorial identities for the number of partitions of a set into subsets of size at least $2$.

scholarly journals Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine and sine functions with applications

2021 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Binomial Coefficients ◽  
Series Expansions ◽  
Bell Polynomials ◽  
Sine Function ◽  
Specific Sequence ◽  
Hyperbolic Cosine ◽  
Faà Di Bruno Formula ◽  
Sine Functions

Abstract In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine (sine) function and the inverse hyperbolic cosine (sine) function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

scholarly journals Some New Classes of Generalized Lagrange-Based Apostol Type Hermite Polynomials

2018 ◽  
Author(s):  
Waseem A. Khan ◽  
K.S. Nisar
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Hermite Polynomials ◽  
Euler Polynomials ◽  
General Symmetry ◽  
Basic Properties ◽  
Genocchi Polynomials ◽  
General Family ◽  
Summation Formulae ◽  
Explicit Representations

In this paper, we introduce a general family of Lagrange-based Apostol-type Hermite polynomials thereby unifying the Lagrange-based Apostol Hermite-Bernoulli and the Lagrange-based Apostol Hermite-Genocchi polynomials. We also define Lagrange-based Apostol Hermite-Euler polynomials via the generating function. In terms of these generalizations, we find new and useful relations between the unified family and the Apostol Hermite-Euler polynomials. We also derive their explicit representations and list some basic properties of each of them. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions.

Sign in / Sign up

Export Citation Format

Share Document