Degenerate Bell polynomials associated with umbral calculus

Abstract Carlitz initiated a study of degenerate Bernoulli and Euler numbers and polynomials which is the pioneering work on degenerate versions of special numbers and polynomials. In recent years, studying degenerate versions regained lively interest of some mathematicians. The purpose of this paper is to study degenerate Bell polynomials by using umbral calculus and generating functions. We derive several properties of the degenerate Bell polynomials including recurrence relations, Dobinski-type formula, and derivatives. In addition, we represent various known families of polynomials such as Euler polynomials, modified degenerate poly-Bernoulli polynomials, degenerate Bernoulli polynomials of the second kind, and falling factorials in terms of degenerate Bell polynomials and vice versa.

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Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials

Mathematics ◽

10.3390/math7010047 ◽

2019 ◽

Vol 7 (1) ◽

pp. 47 ◽

Cited By ~ 4

Author(s):

Taekyun Kim ◽

Dae San Kim

Keyword(s):

Differential Equations ◽

Generating Functions ◽

Bernstein Polynomials ◽

Umbral Calculus ◽

Euler Polynomials ◽

Euler Numbers ◽

Euler Numbers And Polynomials ◽

Special Numbers And Polynomials ◽

Degenerate Bernstein Polynomials ◽

Combinatorial Methods

In recent years, intensive studies on degenerate versions of various special numbers and polynomials have been done by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations. The degenerate Bernstein polynomials and operators were recently introduced as degenerate versions of the classical Bernstein polynomials and operators. Herein, we firstly derive some of their basic properties. Secondly, we explore some properties of the degenerate Euler numbers and polynomials and also their relations with the degenerate Bernstein polynomials.

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New classes of recurrence relations involving hyperbolic functions, special numbers and polynomials

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm201020015s ◽

2021 ◽

pp. 15-15

Author(s):

Yilmaz Simsek

Keyword(s):

Generating Functions ◽

Recurrence Relations ◽

Stirling Numbers ◽

Umbral Calculus ◽

Discrete Mathematics ◽

New Class ◽

Integral Formulas ◽

Euler Numbers And Polynomials ◽

Derivative Formulas ◽

Special Numbers And Polynomials

By using the calculus of finite differences methods and the umbral calculus, we construct recurrence relations for a new class of special numbers. Using this recurrence relation, we define generating functions for this class of special numbers and also new classes of special polynomials. We investigate some properties of these generating functions. By using these generating functions with their functional equations, we obtain many new and interesting identities and relations related to these classes of special numbers and polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers. Finally, some derivative formulas and integral formulas for these classes of special numbers and polynomials are given. In general, this article includes results that have the potential to be used in areas such as discrete mathematics, combinatorics analysis and their applications.

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

AIMS Mathematics ◽

10.3934/math.2022213 ◽

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>

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A Note on a New Type of Degenerate poly-Euler Numbers and Polynomials

10.20944/preprints202007.0202.v1 ◽

2020 ◽

Author(s):

Waseem Khan

Keyword(s):

Arithmetic Function ◽

Bernoulli Numbers ◽

Euler Polynomials ◽

Euler Numbers ◽

Polylogarithm Function ◽

Euler Numbers And Polynomials ◽

Special Numbers And Polynomials ◽

Euler Polynomials And Numbers ◽

New Type ◽

Logarithm Function

Kim-Kim [12] introduced the new type of degenerate Bernoulli numbers and polynomials arising from the degenerate logarithm function. In this paper, we introduce a new type of degenerate poly-Euler polynomials and numbers, are called degenerate poly-Euler polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Euler polynomials and numbers. In the last section, we also consider the degenerate unipoly-Euler polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

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Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials

Symmetry ◽

10.3390/sym12010009 ◽

2019 ◽

Vol 12 (1) ◽

pp. 9 ◽

Cited By ~ 1

Author(s):

Daeyeoul Kim ◽

Yilmaz Simsek ◽

Ji Suk So

Keyword(s):

Generating Functions ◽

Recurrence Relations ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Euler Numbers ◽

Integral Formulas ◽

Euler Numbers And Polynomials ◽

Negative Order ◽

Changhee Polynomials ◽

Changhee Numbers

The purpose of this paper is to construct generating functions for negative order Changhee numbers and polynomials. Using these generating functions with their functional equation, we prove computation formulas for combinatorial numbers and polynomials. These formulas include Euler numbers and polynomials of higher order, Stirling numbers, and negative order Changhee numbers and polynomials. We also give some properties of these numbers and polynomials with their generating functions. Moreover, we give relations among Changhee numbers and polynomials of negative order, combinatorial numbers and polynomials and Bernoulli numbers of the second kind. Finally, we give a partial derivative of an equation for generating functions for Changhee numbers and polynomials of negative order. Using these differential equations, we derive recurrence relations, differential and integral formulas for these numbers and polynomials. We also give p-adic integrals representations for negative order Changhee polynomials including Changhee numbers and Deahee numbers.

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Identities and relations for special numbers and polynomials: An approach to trigonometric functions

Filomat ◽

10.2298/fil2002535k ◽

2020 ◽

Vol 34 (2) ◽

pp. 535-542

Author(s):

Neslihan Kilar ◽

Yilmaz Simsek

Keyword(s):

Generating Functions ◽

Bernoulli Polynomials ◽

Stirling Numbers ◽

Trigonometric Functions ◽

Euler Numbers ◽

Derived Relations ◽

Special Numbers And Polynomials

In this paper, by using trigonometric functions and generating functions, identities and relations associated with special numbers and polynomials are derived. Relations among the combinatorial numbers, the Bernoulli polynomials, the Euler numbers, the Stirling numbers and others special numbers and polynomials are given.

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Hankel determinants of sequences related to Bernoulli and Euler polynomials

International Journal of Number Theory ◽

10.1142/s179304212250021x ◽

2021 ◽

pp. 1-29

Author(s):

Karl Dilcher ◽

Lin Jiu

Keyword(s):

Bernoulli Polynomials ◽

Euler Polynomials ◽

Hankel Determinant ◽

Euler Numbers ◽

Hankel Determinants ◽

Euler Numbers And Polynomials ◽

Dirichlet Characters ◽

Special Cases ◽

Generalized Bernoulli Polynomials ◽

Derivatives Of

We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and polynomials.

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Some Properties for Multiple Twisted (p, q)-L-Function and Carlitz’s Type Higher-Order Twisted (p, q)-Euler Polynomials

Mathematics ◽

10.3390/math7121205 ◽

2019 ◽

Vol 7 (12) ◽

pp. 1205 ◽

Cited By ~ 2

Author(s):

Kyung-Won Hwang ◽

Cheon Seoung Ryoo

Keyword(s):

Generating Functions ◽

Higher Order ◽

Integral Representations ◽

Euler Polynomials ◽

Euler Numbers ◽

Mellin Transformation ◽

Euler Numbers And Polynomials ◽

Symmetric Identities ◽

Positive Integers ◽

Symmetric Property

The main goal of this paper is to study some interesting identities for the multiple twisted ( p , q ) -L-function in a complex field. First, we construct new generating functions of the new Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials. By applying the Mellin transformation to these generating functions, we obtain integral representations of the multiple twisted ( p , q ) -Euler zeta function and multiple twisted ( p , q ) -L-function, which interpolate the Carlitz-type higher order twisted ( p , q ) -Euler numbers and Carlitz-type higher order twisted ( p , q ) -Euler polynomials at non-positive integers, respectively. Second, we get some explicit formulas and properties, which are related to Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials. Third, we give some new symmetric identities for the multiple twisted ( p , q ) -L-function. Furthermore, we also obtain symmetric identities for Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials by using the symmetric property for the multiple twisted ( p , q ) -L-function.

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A ( p , q ) analogue of Poly-Euler polynomials and some related polynomials

Ukrains’kyi Matematychnyi Zhurnal ◽

10.37863/umzh.v72i4.6048 ◽

2020 ◽

Vol 72 (4) ◽

pp. 467-482

Author(s):

T. Komatsu ◽

J. L. Ramírez ◽

V. F. Sirvent

Keyword(s):

Bernoulli Polynomials ◽

Combinatorial Identities ◽

Euler Polynomials ◽

Euler Numbers ◽

Polylogarithm Function ◽

Euler Numbers And Polynomials ◽

Euler Polynomials And Numbers

UDC 517.5 We introduce a ( p , q ) -analogue of the poly-Euler polynomials and numbers by using the ( p , q ) -polylogarithm function. These new sequences are generalizations of the poly-Euler numbers and polynomials. We give several combinatorial identities and properties of these new polynomials, and also show some relations with ( p , q ) -poly-Bernoulli polynomials and ( p , q ) -poly-Cauchy polynomials. The ( p , q ) -analogues generalize the well-known concept of the q -analogue.

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Some recurrence formulas for the q-Bernoulli and q-Euler polynomials

Filomat ◽

10.2298/fil2002663d ◽

2020 ◽

Vol 34 (2) ◽

pp. 663-669

Author(s):

Paçin Dere

Keyword(s):

Recurrence Relations ◽

Bernoulli Polynomials ◽

Umbral Calculus ◽

Euler Polynomials ◽

Appell Polynomials ◽

Derivative Operator ◽

Important Place ◽

Quantum Calculus ◽

Recurrence Formulas ◽

Special Polynomials

The recurrence relations have a very important place for the special polynomials such as q-Appell polynomials. In this paper, we give some recurrence formulas that allow us a better understanding of q-Appell polynomials. We investigate the q-Bernoulli polynomials and q-Euler polynomials, which are q-Appell polynomials, and we obtain their recurrence formulas by using the methods of the q-umbral calculus and the quantum calculus. Our methods include some operators which are quite handy for obtaining relations for the q-Appell polynomials. Especially, some applications of q-derivative operator are used in this work.

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