Degenerate Sheffer sequences and λ-Sheffer sequences

2021 ◽  
Vol 493 (1) ◽  
pp. 124521 ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim
Keyword(s):  
Sheffer Sequences

scholarly journals Finding Determinant Forms of Certain Hybrid Sheffer Sequences

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1105
Author(s):  
Alansari ◽  
Riyasat ◽  
Khan ◽  
Kazmi
Keyword(s):  
Generating Functions ◽  
Integral Transform ◽  
Matrix Approach ◽  
New Family ◽  
Appell Sequences ◽  
Sheffer Sequences

In this article, the integral transform is used to introduce a new family of extended hybrid Sheffer sequences via generating functions and operational rules. The determinant forms and other properties of these sequences are established using a matrix approach. The correspondingresults for the extended hybrid Appell sequences are also obtained. Certain examples in terms of the members of the extended hybrid Sheffer and Appell sequences are framed. By employing operational rules, the identities involving the Lah, Stirling and Pascal matrices are derived for the aforementioned sequences.

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

Self-Inverse Sheffer Sequences

1976 ◽  
Vol 7 (5) ◽  
pp. 723-728 ◽  
Author(s):  
James Ward Brown ◽  
Marek Kuczma
Keyword(s):  
Sheffer Sequences

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 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 Powers under umbral composition and degeneration for Sheffer sequences

2016 ◽  
Vol 2016 (1) ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim ◽  
Hyuck In Kwon ◽  
Toufik Mansour
Keyword(s):  
Sheffer Sequences

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 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.

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