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

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.

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Some Properties of $Q$-Hermite Fubini Numbers and Polynomials

10.20944/preprints201904.0309.v1 ◽

2019 ◽

Author(s):

Waseem A. Khan

Keyword(s):

Generating Functions ◽

Hermite Polynomials ◽

Bernoulli Polynomials ◽

Stirling Numbers ◽

Euler Polynomials ◽

New Class ◽

Genocchi Polynomials ◽

Summation Formulas

The main purpose of this paper is to introduce a new class of $q$-Hermite-Fubini numbers and polynomials by combining the $q$-Hermite polynomials and $q$-Fubini polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive $q$-integers.  Also, we establish some relationships for $q$-Hermite-Fubini polynomials associated with $q$-Bernoulli polynomials, $q$-Euler polynomials and $q$-Genocchi polynomials and $q$-Stirling numbers of the second kind.

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A Note on Type 2 Degenerate Multi-Poly-Bernoulli Polynomials and Numbers

10.20944/preprints202008.0057.v1 ◽

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.

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On generating functions for the special polynomials

Filomat ◽

10.2298/fil1701009s ◽

2017 ◽

Vol 31 (1) ◽

pp. 9-16 ◽

Cited By ~ 2

Author(s):

Yilmaz Simsek

Keyword(s):

Generating Functions ◽

Functional Equations ◽

Bernoulli Polynomials ◽

Stirling Numbers ◽

Special Polynomials ◽

New Family ◽

Bernoulli Polynomials And Numbers ◽

Computation Algorithm ◽

Generalized Stirling Numbers

The aim of this paper is to investigate and give a new family of Apell type polynomials, which are related to the Euler, Frobenius-Euler and Apostol-Bernoulli polynomials and numbers and also the generalized Stirling numbers of the second kind etc. The results presented in this paper are based upon the theory of the generating functions. By using functional equations of these generating functions, we drive some identities and relations for these numbers and polynomials. Moreover, we give a computation algorithm these numbers.

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A Note on $(P,Q)$-Analogue Type of Fubini Numbers and Polynomials

10.20944/preprints201904.0308.v1 ◽

2019 ◽

Author(s):

Waseem A. Khan

Keyword(s):

Bernoulli Polynomials ◽

Stirling Numbers ◽

Euler Polynomials ◽

New Class ◽

Genocchi Polynomials ◽

Summation Formulas

In this paper, we introduce a new class of  $(p,q)$-analogue type of Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we consider some relationships for  $(p,q)$-Fubini polynomials associated with $(p,q)$-Bernoulli polynomials, $(p,q)$-Euler polynomials and $(p,q)$-Genocchi polynomials and $(p,q)$-Stirling numbers of the second kind.

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Two-Variable Type 2 Poly-Fubini Polynomials

Mathematics ◽

10.3390/math9030281 ◽

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.

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Some Identities on the Poly-Genocchi Polynomials and Numbers

Symmetry ◽

10.3390/sym12061007 ◽

2020 ◽

Vol 12 (6) ◽

pp. 1007 ◽

Cited By ~ 4

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.

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Multifarious Results for q-Hermite Based Frobenius Type Eulerian Polynomials

10.20944/preprints201908.0215.v1 ◽

2019 ◽

Author(s):

Waseem Khan ◽

Idrees Ahmad Khan ◽

Mehmet Acikgoz ◽

Ugur Duran

Keyword(s):

Generating Function ◽

Generating Functions ◽

Recurrence Relations ◽

Series Representation ◽

Bernoulli Polynomials ◽

Stirling Numbers ◽

Eulerian Polynomials ◽

New Class ◽

Genocchi Polynomials

In this paper, a new class of q-Hermite based Frobenius type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via different generating methods. Furthermore, diverse correlations including the q-Apostol-Bernoulli polynomials, the q-Apostol-Euler poynoomials, the q-Apostol-Genocchi polynomials and the q-Stirling numbers of the second kind are also established by means of the their generating functions.

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New degenerate Bernoulli, Euler, and Genocchi polynomials

Pure Mathematics and Applications ◽

10.1515/puma-2015-0037 ◽

2020 ◽

Vol 29 (1) ◽

pp. 1-16

Author(s):

Orli Herscovici ◽

Toufik Mansour

Keyword(s):

Exponential Function ◽

Bernoulli Polynomials ◽

Genocchi Polynomials ◽

Bernoulli Polynomials And Numbers ◽

Euler Identity

AbstractWe introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function. Also, we present generalizations of some familiar identities and connection between these types of Bernoulli, Euler, and Genocchi polynomials. Moreover, we establish new analogues of the Euler identity for degenerate Bernoulli polynomials and numbers.

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Formulas involving sums of powers, special numbers and polynomials arising from p-adic integrals, trigonometric and generating functions

Publications de l Institut Mathematique ◽

10.2298/pim2022103k ◽

2020 ◽

Vol 108 (122) ◽

pp. 103-120

Author(s):

Neslihan Kilar ◽

Yilmaz Simsek

Keyword(s):

Generating Functions ◽

Bernoulli Polynomials ◽

Stirling Numbers ◽

Sums Of Powers ◽

Alternating Sums ◽

Special Numbers And Polynomials ◽

Bernoulli Polynomials And Numbers ◽

Euler Polynomials And Numbers ◽

Historical Remarks ◽

Positive Integers

The formula for the sums of powers of positive integers, given by Faulhaber in 1631, is proven by using trigonometric identities and some properties of the Bernoulli polynomials. Using trigonometric functions identities and generating functions for some well-known special numbers and polynomials, many novel formulas and relations including alternating sums of powers of positive integers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Fubini numbers, the Stirling numbers, the tangent numbers are also given. Moreover, by applying the Riemann integral and p-adic integrals involving the fermionic p-adic integral and the Volkenborn integral, some new identities and combinatorial sums related to the aforementioned numbers and polynomials are derived. Furthermore, we serve up some revealing and historical remarks and observations on the results of this paper.

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Bell-Based Bernoulli Polynomials with Applications

Axioms ◽

10.3390/axioms10010029 ◽

2021 ◽

Vol 10 (1) ◽

pp. 29

Author(s):

Ugur Duran ◽

Serkan Araci ◽

Mehmet Acikgoz

Keyword(s):

Bernoulli Polynomials ◽

Stirling Numbers ◽

Umbral Calculus ◽

Bell Polynomials ◽

Summation Formulas ◽

Symmetric Identities

In this paper, we consider Bell-based Stirling polynomials of the second kind and derive some useful relations and properties including some summation formulas related to the Bell polynomials and Stirling numbers of the second kind. Then, we introduce Bell-based Bernoulli polynomials of order α and investigate multifarious correlations and formulas including some summation formulas and derivative properties. Also, we acquire diverse implicit summation formulas and symmetric identities for Bell-based Bernoulli polynomials of order α. Moreover, we attain several interesting formulas of Bell-based Bernoulli polynomials of order α arising from umbral calculus.

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