scholarly journals A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties

2021 ◽  
Vol 6 (11) ◽  
pp. 12680-12697
Author(s):  
Waseem A. Khan ◽  
◽  
Abdulghani Muhyi ◽  
Rifaqat Ali ◽  
Khaled Ahmad Hassan Alzobydi ◽  
...  
Keyword(s):  
Bernoulli Polynomials ◽  
Graphical Representations ◽  
New Family ◽  
Special Numbers And Polynomials ◽  
Explicit Expressions

<abstract><p>The main object of this article is to present type 2 degenerate poly-Bernoulli polynomials of the second kind and numbers by arising from modified degenerate polyexponential function and investigate some properties of them. Thereafter, we treat the type 2 degenerate unipoly-Bernoulli polynomials of the second kind via modified degenerate polyexponential function and derive several properties of these polynomials. Furthermore, some new identities and explicit expressions for degenerate unipoly polynomials related to special numbers and polynomials are obtained. In addition, certain related beautiful zeros and graphical representations are displayed with the help of <italic>Mathematica</italic>.</p></abstract>

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 Analytical properties of type 2 degenerate poly-Bernoulli polynomials associated with their applications

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Waseem A. Khan ◽  
Ghulam Muhiuddin ◽  
Abdulghani Muhyi ◽  
Deena Al-Kadi
Keyword(s):  
Arithmetic Function ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Analytical Properties ◽  
Special Numbers And Polynomials ◽  
Bernoulli Polynomials And Numbers ◽  
Degenerate Type ◽  
Explicit Expressions

AbstractRecently, Kim et al. (Adv. Differ. Equ. 2020:168, 2020) considered the poly-Bernoulli numbers and polynomials resulting from the moderated version of degenerate polyexponential functions. In this paper, we investigate the degenerate type 2 poly-Bernoulli numbers and polynomials which are derived from the moderated version of degenerate polyexponential functions. Our degenerate type 2 degenerate poly-Bernoulli numbers and polynomials are different from those of Kim et al. (Adv. Differ. Equ. 2020:168, 2020) and Kim and Kim (Russ. J. Math. Phys. 26(1):40–49, 2019). Utilizing the properties of moderated degenerate poly-exponential function, we explore some properties of our type 2 degenerate poly-Bernoulli numbers and polynomials. From our investigation, we derive some explicit expressions for type 2 degenerate poly-Bernoulli numbers and polynomials. In addition, we also scrutinize type 2 degenerate unipoly-Bernoulli polynomials related to an arithmetic function and investigate some identities for those polynomials. In particular, we consider certain new explicit expressions and relations of type 2 degenerate unipoly-Bernoulli polynomials and numbers related to special numbers and polynomials. Further, some related beautiful zeros and graphical representations are displayed with the help of Mathematica.

scholarly journals A Note on Type 2 Degenerate Poly-Fubini Polynomials and Numbers

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

In this paper, we construct the degenerate poly-Fubini polynomials, called the type 2 degenerate poly-Fubini polynomials, by using the modified degenerate polyexponential function and derive several properties on the degenerate poly-Fubini polynomials and numbers. In the last section, we introduce type 2 degenerate unipoly- Fubini polynomials attached to an arithmetic function, by using the modified degenerate polyexponential function and investigate some identities for those polynomials. Furthermore, 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 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.

scholarly journals Two-Variable Type 2 Poly-Fubini Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 281
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran
Keyword(s):  
Integral Representation ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Higher Order ◽  
Variable Type ◽  
Summation Formulas

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the first kinds, and the Daehee polynomials are acquired.

scholarly journals Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion: Part II: For confluent family of hypergeometric equations

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Kohei Iwaki ◽  
Tatsuya Koike ◽  
Yumiko Takei
Keyword(s):  
Free Energy ◽  
Differential Equations ◽  
Classical Limit ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Spectral Curves ◽  
Topological Recursion ◽  
Hypergeometric Equations ◽  
Explicit Expressions

Abstract We show that each member of the confluent family of the Gauss hypergeometric equations is realized as quantum curves for appropriate spectral curves. As an application, relations between the Voros coefficients of those equations and the free energy of their classical limit computed by the topological recursion are established. We will also find explicit expressions of the free energy and the Voros coefficients in terms of the Bernoulli numbers and Bernoulli polynomials. Communicated by: Youjin Zhang

scholarly journals Some Identities on the Poly-Genocchi Polynomials and Numbers

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1007 ◽  
Author(s):  
Dmitry V. Dolgy ◽  
Lee-Chae Jang
Keyword(s):  
Arithmetic Function ◽  
Bernoulli Polynomials ◽  
Genocchi Polynomials

Recently, Kim-Kim (2019) introduced polyexponential and unipoly functions. By using these functions, they defined type 2 poly-Bernoulli and type 2 unipoly-Bernoulli polynomials and obtained some interesting properties of them. Motivated by the latter, in this paper, we construct the poly-Genocchi polynomials and derive various properties of them. Furthermore, we define unipoly Genocchi polynomials attached to an arithmetic function and investigate some identities of them.

scholarly journals A Note on Type 2 Degenerate Poly-Frobenius-Euler Polynomials

2020 ◽  
Author(s):  
Waseem Khan
Keyword(s):  
Euler Polynomials ◽  
Genocchi Polynomials ◽  
Explicit Expressions

In this paper, we construct the degenerate poly-Frobenius-Genocchi polynomials, called the type 2 degenerate poly-Frobenius-Euler polynomials, by means of polyexponential function. We derive explicit expressions and some identities of those polynomials. In the last section, we introduce type 2 degenerate unipoly-Frobenius-Genocchi polynomials by means of unipoly function and derive explicit multifarious properties.

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