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 Construction of the Type 2 Poly-Frobenius-Genocchi Polynomials with Their Certain Applications

2020 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Genocchi Polynomials ◽  
Definition Of ◽  
Special Case

Motivated by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim, in the present paper, we consider a class of new generating function for the Frobenius-Genocchi polynomials, called the type 2 poly-Frobenius-Genocchi polynomials, by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Frobenius-Genocchi polynomias equal a linear combination of the classical Frobenius-Genocchi polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Frobenius-Genocchi polynomials and Bernoulli polynomials of order k. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim, we introduce the unipoly-Frobenius-Genocchi polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Frobenius-Genocchi polynomials and the classical Frobenius-Genocchi polynomials.

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

2020 ◽  
Author(s):  
Waseem A. Khan
Keyword(s):  
Linear Combination ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Genocchi Polynomials

In [18], Kim et al. introduced the degenerate poly-Bernoulli polynomials by using polyexponential function. In this paper, we study the degenerate poly-Frobenius-Genocchi polynomials, which are called the type 2 degenerate poly-Frobenius-Genocchi polynomials, by means of polyexponential function. Then, we derive some useful relations and properties. We derive type 2 degenerate poly-Frobenius-Genocchi polynomials equal a linear combination of the degenerate Frobenius-Genocchi polynomials and Stirling numbers of the first kind. Furthermore, we introduce type 2 degenerate unipoly-Frobenius-Genocchi polynomials by means of unipoly function and derive explicit multifarious properties.

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 Multi-Poly-Bernoulli Polynomials and Numbers

2020 ◽  
Author(s):  
Waseem A Khan ◽  
Aysha Khan ◽  
Ugur Duran
Keyword(s):  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Addition Formula ◽  
Genocchi Polynomials ◽  
Whitney Numbers ◽  
Derivative Formula ◽  
Bernoulli Polynomials And Numbers ◽  
Definition Of ◽  
Special Case

Inspired by the definition of degenerate multi-poly-Genocchi polynomials given by using the degenerate multi-polyexponential functions. In this paper, we consider a class of new generating function for the degenerate multi-poly-Bernoulli polynomials, called the type 2 degenerate multi-poly-Bernoulli polynomials by means of the degenerate multiple polyexponential functions. Then, we investigate their some properties and relations. We show that the type 2 degenerate multi-poly-Bernoulli polynomials equals a linear combination of the weighted degenerate Bernoulli polynomials and Stirling numbers of the first kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Bernoulli numbers and degenerate Whitney numbers.

scholarly journals A New Family of Frobenius-Genocchi Polynomials and Its Certain Properties

2020 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Arithmetic Function ◽  
Hermite Polynomials ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Genocchi Numbers ◽  
Genocchi Polynomials ◽  
New Family ◽  
Summation Formulas ◽  
Derivative Formula ◽  
Bernoulli Polynomials And Numbers

Recently, Kim-Kim [13] have introduced polyexponential functions as an inverse to the polylogarithm functions, and constructed type 2 poly-Bernoulli polynomials. They have also introduced unipoly functions attached to each suitable arithmetic function as a universal concept. Inspired by their work, in this paper, we introduce a new class of the Frobenius-Genocchi polynomials. We derive the diverse formulas and identities covering some summation formulas, derivative formula and correlations with Bernoulli polynomials and numbers, Stirling numbers of the both kinds, degenerate Frobenius-Genocchi polynomials and degenerate Frobenius-Euler polynomials. Moreover, by using the unipoly function as following Kim-Kim's work in <cite>Kim1</cite>, we consider degenerate unipoly-Frobenius-Genocchi polynomials and investigate some formulas and relationships with Daehee numbers, degenerate Frobenius-Genocchi numbers and Stirling numbers of the first kind. Finally, we obtain an Gaussian integral representation of the Frobenius-Genocchi polynomials in terms of the 2-variable Hermite 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 Some Identities on the Generalizedq-Bernoulli,q-Euler, andq-Genocchi Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Daeyeoul Kim ◽  
Burak Kurt ◽  
Veli Kurt
Keyword(s):  
Recurrence Relations ◽  
Analogous Result ◽  
Bernoulli Polynomials ◽  
Addition Theorem ◽  
Euler Polynomials ◽  
Genocchi Polynomials

Mahmudov (2012, 2013) introduced and investigated someq-extensions of theq-Bernoulli polynomialsℬn,qαx,yof orderα, theq-Euler polynomialsℰn,qαx,yof orderα, and theq-Genocchi polynomials𝒢n,qαx,yof orderα. In this paper, we give some identities forℬn,qαx,y,𝒢n,qαx,y, andℰn,qαx,yand the recurrence relations between these polynomials. This is an analogous result to theq-extension of the Srivastava-Pintér addition theorem in Mahmudov (2013).

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