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 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 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 Properties of certain new special polynomials associated with Sheffer sequences

2016 ◽  
Vol 9 (1) ◽  
pp. 245-270
Author(s):  
Nusrat Raza ◽  
Subuhi Khan ◽  
Mahvish Ali
Keyword(s):  
Special Polynomials ◽  
Sheffer Sequences

scholarly journals An algebraic exposition of umbral calculus with application to general linear interpolation problem: A survey

2014 ◽  
Vol 96 (110) ◽  
pp. 67-83 ◽  
Author(s):  
Francesco Costabile ◽  
Elisabetta Longo
Keyword(s):  
Interpolation Problem ◽  
Linear Interpolation ◽  
Umbral Calculus ◽  
General Linear ◽  
Numerical Examples ◽  
Polynomial Sequences ◽  
Sheffer Sequences ◽  
Determinantal Form ◽  
Systematic Exposition ◽  
Sheffer Polynomial

A systematic exposition of Sheffer polynomial sequences via determinantal form is given. A general linear interpolation problem related to Sheffer sequences is considered. It generalizes many known cases of linear interpolation. Numerical examples and conclusions close the paper.

scholarly journals Legendre-Gould Hopper Based Sheffer Polynomials: Properties and Applications

2020 ◽  
Author(s):  
Ghazala Yasmin ◽  
Abdulghani Muhyi
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Mixed Type ◽  
Integral Representations ◽  
Special Polynomials ◽  
Sheffer Sequences ◽  
Sheffer Polynomials

In this article, the Legendre-Gould Hopper polynomials are combined with Sheffer sequences to introduce certain mixed type special polynomials. Generating functions, differential equations and certain other properties of Legendre-Gould Hopper based Sheffer polynomials are derived. Further, operational and integral representations providing connections between these polynomials and known special polynomials are established. Certain identities and results for some members of these new mixed polynomials are also obtained. Finally, the determinantal definitions of Legendre-Gould Hopper based Sheffer polynomials are also given.

scholarly journals On differential polynomials I

1997 ◽  
Vol 148 ◽  
pp. 39-72 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Differential Polynomials ◽  
Polynomial Sequences ◽  
Special Polynomials

AbstractThe content of Part I is nothing else than, the theory of binomial polynomial sequences in infinite variables (u(1), u(2), u(3), …) with weight u(1) = l. However, sometimes we are concerned with specialization therefore, we call the elements in K[u(1), u(2), u(3), …] differential polynomials. As analogies of special polynomials with binomial property, we may construct special differential polynomials with binomial property.

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 The Mean Values of Character Sums and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 318
Author(s):  
Jiafan Zhang ◽  
Yuanyuan Meng
Keyword(s):  
Character Sums ◽  
Gauss Sums ◽  
Mean Values ◽  
Special Polynomials ◽  
Elementary Methods ◽  
The Mean

In this paper, we use the elementary methods and properties of classical Gauss sums to study the calculation problems of some mean values of character sums of special polynomials, and obtained several interesting calculation formulae for them. As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.

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 the speed of convergence of Newton’s method for complex polynomials

2015 ◽  
Vol 85 (298) ◽  
pp. 693-705 ◽  
Author(s):  
Todor Bilarev ◽  
Magnus Aspenberg ◽  
Dierk Schleicher
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Speed Of Convergence ◽  
Complex Polynomials
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