scholarly journals New Type of Degenerate Poly-Frobenius-Euler Polynomials and Numbers

2020 ◽  
Author(s):  
Waseem Khan
Keyword(s):  
Bernoulli Polynomials ◽  
Euler Polynomials ◽  
Polylogarithm Function ◽  
Euler Polynomials And Numbers ◽  
New Type ◽  
Explicit Expressions

Motivated by Kim-Kim [19] introduced the new type of degenerate poly- Bernoulli polynomials by means of the degenerate polylogarithm function. In this paper, we define the degenerate poly-Frobenius-Euler polynomials, called the new type of degenerate poly-Frobenius-Euler polynomials, by means of the degenerate polylogarithm function. Then, we derive explicit expressions and some identities of those numbers and polynomials.

scholarly journals Type 2 Degenerate Poly-Euler Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1011 ◽  
Author(s):  
Dae Sik Lee ◽  
Hye Kyung Kim ◽  
Lee-Chae Jang
Keyword(s):  
Final Section ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
Euler Polynomials And Numbers ◽  
New Type ◽  
Explicit Expressions

In recent years, many mathematicians have studied the degenerate versions of many special polynomials and numbers. The polyexponential functions were introduced by Hardy and rediscovered by Kim, as inverses to the polylogarithms functions. The paper is divided two parts. First, we introduce a new type of the type 2 poly-Euler polynomials and numbers constructed from the modified polyexponential function, the so-called type 2 poly-Euler polynomials and numbers. We show various expressions and identities for these polynomials and numbers. Some of them involving the (poly) Euler polynomials and another special numbers and polynomials such as (poly) Bernoulli polynomials, the Stirling numbers of the first kind, the Stirling numbers of the second kind, etc. In final section, we introduce a new type of the type 2 degenerate poly-Euler polynomials and the numbers defined in the previous section. We give explicit expressions and identities involving those polynomials in a similar direction to the previous section.

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

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.

scholarly journals On type 2 degenerate Bernoulli and Euler polynomials of complex variable

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Lee-Chae Jang ◽  
Han-Young Kim
Keyword(s):  
Complex Variable ◽  
Bernoulli Polynomials ◽  
Affirmative Answer ◽  
Euler Polynomials ◽  
The Real ◽  
New Type ◽  
Explicit Expressions

AbstractRecently, Masjed-Jamei, Beyki, and Koepf studied the so-called new type Euler polynomials without using Euler polynomials of complex variable. Here we study the type 2 degenerate cosine-Euler and type 2 degenerate sine-Euler polynomials, which are type 2 degenerate versions of these new type Euler polynomials, by considering the degenerate Euler polynomials of complex variable and by treating the real and imaginary parts separately. In addition, we investigate the corresponding ones for Bernoulli polynomials in the same manner. We derive some explicit expressions for those new polynomials and some identities relating to them. Here we note that the idea of separating the real and imaginary parts separately gives an affirmative answer to the question asked by Hacène Belbachir.

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

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.

scholarly journals On A New Type of Degenerate Poly-Fubini Numbers and Polynomials

2020 ◽  
Author(s):  
Waseem Khan
Keyword(s):  
Arithmetic Function ◽  
Polylogarithm Function ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Explicit Expressions

In this paper, we introduce a new type of degenerate poly-Fubini polynomials and numbers, are called degenerate poly-Fubini polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Fubini polynomials and numbers. In the last section, we also consider the degenerate unipoly-Fubini 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.

scholarly journals Note on the Type 2 Degenerate Multi-Poly-Euler Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1691
Author(s):  
Waseem Ahmad Khan ◽  
Mehmet Acikgoz ◽  
Ugur Duran
Keyword(s):  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Addition Formula ◽  
Whitney Numbers ◽  
Derivative Formula ◽  
New Type ◽  
Special Case ◽  
Math Phys

Kim and Kim (Russ. J. Math. Phys. 26, 2019, 40-49) introduced polyexponential function as an inverse to the polylogarithm function and by this, constructed a new type poly-Bernoulli polynomials. Recently, by using the polyexponential function, a number of generalizations of some polynomials and numbers have been presented and investigated. Motivated by these researches, in this paper, multi-poly-Euler polynomials are considered utilizing the degenerate multiple polyexponential functions and then, their properties and relations are investigated and studied. That the type 2 degenerate multi-poly-Euler polynomials equal a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind is proved. Moreover, an addition formula and a derivative formula are derived. Furthermore, in a special case, a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers is shown.

scholarly journals A Note on Degenerate Euler and Bernoulli Polynomials of Complex Variable

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1168 ◽  
Author(s):  
Dae Kim ◽  
Taekyun Kim ◽  
Hyunseok Lee
Keyword(s):  
Complex Variable ◽  
Bernoulli Polynomials ◽  
Affirmative Answer ◽  
Euler Polynomials ◽  
New Type

Recently, the so-called new type Euler polynomials have been studied without considering Euler polynomials of a complex variable. Here we study degenerate versions of these new type Euler polynomials. This has been done by considering the degenerate Euler polynomials of a complex variable. We also investigate corresponding ones for Bernoulli polynomials in the same manner. We derive some properties and identities for those new polynomials. Here we note that our result gives an affirmative answer to the question raised by the reviewer of the paper.

q-Analogs of Lidstone expansion theorem, two-point Taylor expansion theorem, and Bernoulli polynomials

2019 ◽  
Vol 17 (06) ◽  
pp. 853-895
Author(s):  
Mourad E. H. Ismail ◽  
Zeinab S. I. Mansour
Keyword(s):  
Taylor Expansion ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Bernoulli Polynomials ◽  
Classical Case ◽  
The Other ◽  
Euler Polynomials ◽  
Expansion Theorem ◽  
Bernoulli Polynomials And Numbers ◽  
Euler Polynomials And Numbers

In this paper, we introduce a generalization of the [Formula: see text]-Taylor expansion theorems. We expand a function in a neighborhood of two points instead of one in three different theorems. The first is a [Formula: see text]-analog of the Lidstone theorem where the two points are 0 and 1 and we expand the function in [Formula: see text]-analogs of Lidstone polynomials which are in fact [Formula: see text]-Bernoulli polynomials as in the classical case. The definitions of these [Formula: see text]-Bernoulli polynomials and numbers are introduced. We also introduce [Formula: see text]-analogs of Euler polynomials and numbers. On the other two expansion theorems, we expand an analytic function around arbitrary points [Formula: see text] and [Formula: see text] either in terms of the polynomials [Formula: see text] or in terms of the polynomials [Formula: see text]. As an application, we introduce a new series expansion for the basic hypergeometric series [Formula: see text].

scholarly journals A New Family of Degenerate Poly-Genocchi Polynomials with Its Certain Properties

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Waseem A. Khan ◽  
Rifaqat Ali ◽  
Khaled Ahmad Hassan Alzobydi ◽  
Naeem Ahmed
Keyword(s):  
Arithmetic Function ◽  
Polylogarithm Function ◽  
Genocchi Polynomials ◽  
New Family ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Explicit Expressions

In this paper, we introduce a new type of degenerate Genocchi polynomials and numbers, which are called degenerate poly-Genocchi polynomials and numbers, by using the degenerate polylogarithm function, and we derive several properties of these polynomials systematically. Then, we also consider the degenerate unipoly-Genocchi 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.

A New Type of Euler Polynomials and Numbers

2018 ◽  
Vol 15 (3) ◽  
Author(s):  
M. Masjed-Jamei ◽  
M. R. Beyki ◽  
W. Koepf
Keyword(s):  
Euler Polynomials ◽  
Euler Polynomials And Numbers ◽  
New Type
Sign in / Sign up

Export Citation Format

Share Document