scholarly journals Some Identities of the Degenerate Higher Order Derangement Polynomials and Numbers

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 176
Author(s):  
Hye Kyung Kim
Keyword(s):  
Higher Order ◽  
Umbral Calculus ◽  
Inversion Formulas ◽  
Special Polynomials ◽  
Sheffer Sequence ◽  
Sheffer Sequences

Recently, Kim-Kim (J. Math. Anal. Appl. (2021), Vol. 493(1), 124521) introduced the λ-Sheffer sequence and the degenerate Sheffer sequence. In addition, Kim et al. (arXiv:2011.08535v1 17 November 2020) studied the degenerate derangement polynomials and numbers, and investigated some properties of those polynomials without using degenerate umbral calculus. In this paper, the y the degenerate derangement polynomials of order s (s∈N) and give a combinatorial meaning about higher order derangement numbers. In addition, the author gives some interesting identities related to the degenerate derangement polynomials of order s and special polynomials and numbers by using degenerate Sheffer sequences, and at the same time derive the inversion formulas of these identities.

scholarly journals Degenerate Lah–Bell polynomials arising from degenerate Sheffer sequences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Hye Kyung Kim
Keyword(s):  
Differential Operators ◽  
Umbral Calculus ◽  
Bell Polynomials ◽  
Linear Functionals ◽  
Inversion Formulas ◽  
Sheffer Sequence ◽  
Sheffer Sequences ◽  
Special Numbers And Polynomials ◽  
Symmetric Identities ◽  
Degenerate Version

AbstractUmbral calculus is one of the important methods for obtaining the symmetric identities for the degenerate version of special numbers and polynomials. Recently, Kim–Kim (J. Math. Anal. Appl. 493(1):124521, 2021) introduced the λ-Sheffer sequence and the degenerate Sheffer sequence. They defined the λ-linear functionals and λ-differential operators, respectively, instead of the linear functionals and the differential operators of umbral calculus established by Rota. In this paper, the author gives various interesting identities related to the degenerate Lah–Bell polynomials and special polynomials and numbers by using degenerate Sheffer sequences, and at the same time derives the inversion formulas of these identities.

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 On the type 2 poly-Bernoulli polynomials associated with umbral calculus

2021 ◽  
Vol 19 (1) ◽  
pp. 878-887
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Jin-Woo Park
Keyword(s):  
Bernoulli Polynomials ◽  
Higher Order ◽  
Umbral Calculus ◽  
Euler Polynomials ◽  
Special Polynomials

Abstract Type 2 poly-Bernoulli polynomials were introduced recently with the help of modified polyexponential functions. In this paper, we investigate several properties and identities associated with those polynomials arising from umbral calculus techniques. In particular, we express the type 2 poly-Bernoulli polynomials in terms of several special polynomials, like higher-order Cauchy polynomials, higher-order Euler polynomials, and higher-order Frobenius-Euler polynomials.

scholarly journals Voigt Transform and Umbral Image

2020 ◽  
Vol 25 (3) ◽  
pp. 49
Author(s):  
Silvia Licciardi ◽  
Rosa Maria Pidatella ◽  
Marcello Artioli ◽  
Giuseppe Dattoli
Keyword(s):  
Spectroscopic Studies ◽  
Higher Order ◽  
Point Of View ◽  
The Other ◽  
Pivotal Role ◽  
Umbral Calculus ◽  
Laguerre Functions ◽  
Hermite And Laguerre Functions ◽  
Higher Order Functions ◽  
Voigt Function

In this paper, we show that the use of methods of an operational nature, such as umbral calculus, allows achieving a double target: on one side, the study of the Voigt function, which plays a pivotal role in spectroscopic studies and in other applications, according to a new point of view, and on the other, the introduction of a Voigt transform and its possible use. Furthermore, by the same method, we point out that the Hermite and Laguerre functions, extension of the corresponding polynomials to negative and/or real indices, can be expressed through a definition in a straightforward and unified fashion. It is illustrated how the techniques that we are going to suggest provide an easy derivation of the relevant properties along with generalizations to higher order functions.

scholarly journals A note on the higher-order Frobenius-Euler polynomials and Sheffer sequences

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim ◽  
Sang-Hun Lee ◽  
Seog-Hoon Rim
Keyword(s):  
Higher Order ◽  
Euler Polynomials ◽  
Sheffer Sequences

scholarly journals Simplification of Coefficients in Differential Equations Associated with Higher Order Frobenius–Euler Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stirling Numbers ◽  
Higher Order ◽  
Bell Polynomials ◽  
Euler Numbers ◽  
Inversion Formulas

In the paper, by virtue of the Fa`a di Bruno formula, some properties of the Bell polynomials of the second kind, and the inversion formulas of binomial numbers and the Stirling numbers of the first and second kinds, the authors simplify meaningfully and significantly coefficients in two families of ordinary differential equations associated with higher order Frobenius&ndash;Euler numbers.

scholarly journals A Class of Sheffer Sequences of Some Complex Polynomials and Their Degenerate Types

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1064 ◽  
Author(s):  
Dojin Kim
Keyword(s):  
Trigonometric Polynomial ◽  
Complex Polynomials ◽  
Polynomial Sequences ◽  
Special Polynomials ◽  
Sheffer Sequences

We study some properties of Sheffer sequences for some special polynomials with complex Changhee and Daehee polynomials introducing their complex versions of the polynomials and splitting them into real and imaginary parts using trigonometric polynomial sequences. Moreover, considering their degenerate types of Sheffer sequences based on umbral composition, we present some useful expressions, properties, and examples about complex versions of the degenerate polynomials.

scholarly journals A Note on Type 2 w-Daehee Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 697
Author(s):  
Minyoung Ma ◽  
Dongkyu Lim
Keyword(s):  
Order Type ◽  
Higher Order ◽  
Special Polynomials ◽  
Invariant Integral ◽  
Symmetric Identities

In the paper, by virtue of the p-adic invariant integral on Z p , the authors consider a type 2 w-Daehee polynomials and present some properties and identities of these polynomials related with well-known special polynomials. In addition, we present some symmetric identities involving the higher order type 2 w-Daehee polynomials. These identities extend and generalize some known results.

scholarly journals Dual Numbers and Operational Umbral Methods

Axioms ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 77 ◽  
Author(s):  
Nicolas Behr ◽  
Giuseppe Dattoli ◽  
Ambra Lattanzi ◽  
Silvia Licciardi
Keyword(s):  
Finite Difference ◽  
Higher Order ◽  
Point Of View ◽  
Umbral Calculus ◽  
Dual Numbers ◽  
Numerical Computations

Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus.

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