scholarly journals A Note on the Truncated-Exponential Based Apostol-Type Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 538 ◽  
Author(s):  
H. M. Srivastava ◽  
Serkan Araci ◽  
Waseem A. Khan ◽  
Mehmet Acikgöz
Keyword(s):  
Generating Functions ◽  
Bernoulli Polynomials ◽  
New Family ◽  
Summation Formulas ◽  
Symmetric Identities

In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric identities by using their generating functions. The results, which we have derived here, provide generalizations of the corresponding known formulas including identities involving generalized Hermite-Bernoulli polynomials.

scholarly journals A New Class of Hermite-Fubini Polynomials and Its Properties

2019 ◽  
Author(s):  
Waseem Khan ◽  
Nisar K S
Keyword(s):  
Generating Functions ◽  
New Class ◽  
Summation Formulas ◽  
Symmetric Identities

In this paper, we introduce a new class of Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of Hermite-Fubini numbers and polynomials by using generating functions.

scholarly journals Bell-Based Bernoulli Polynomials with Applications

Axioms ◽  
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.

scholarly journals A Note on Degenerate Hermite-Fubini Numbers and Polynomials

2019 ◽  
Author(s):  
Waseem A. Khan ◽  
Idrees A. Khan ◽  
Nisar K S
Keyword(s):  
Generating Functions ◽  
New Class ◽  
Summation Formulas ◽  
Symmetric Identities

In this paper, we introduce a new class of degenerate Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of degenerate Hermite-Fubini numbers and polynomials by using generating functions.

scholarly journals Certain Properties of Apostol-Type Hermite-Based- Frobenius-Genocchi Polynomials

2021 ◽  
Vol 45 (6) ◽  
pp. 859-872
Author(s):  
WASEEM A. KHAN ◽  
◽  
DIVESH SRIVASTAVA
Keyword(s):  
Generating Functions ◽  
Bernoulli Polynomials ◽  
Array Type ◽  
Genocchi Polynomials ◽  
Summation Formulae ◽  
Symmetric Identities ◽  
Set Up

This paper is well designed to set-up some new identities related to generalized Apostol-type Hermite-based-Frobenius-Genocchi polynomials and by applying the generating functions, we derive some implicit summation formulae and symmetric identities. Further a relationship between Array-type polynomials, Apostol-type Bernoulli polynomials and generalized Apostol-type Frobenius-Genocchi polynomials is also established.

scholarly journals On the generalized Apostol-type Frobenius-Genocchi polynomials

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1967-1977 ◽  
Author(s):  
Waseem Khan ◽  
Divesh Srivastava
Keyword(s):  
Generating Functions ◽  
Bernoulli Polynomials ◽  
Array Type ◽  
New Class ◽  
Genocchi Polynomials ◽  
Summation Formulae ◽  
Symmetric Identities

The main object of this work is to introduce a new class of the generalized Apostol-type Frobenius-Genocchi polynomials and is to investigate some properties and relations of them. We derive implicit summation formulae and symmetric identities by applying the generating functions. In addition a relation in between Array-type polynomials, Apostol-Bernoulli polynomials and generalized Apostol-type Frobenius-Genocchi polynomials is also given.

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

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.

On generating functions for the special polynomials

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 9-16 ◽  
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.

scholarly journals A New Class of Laguerre-Based Hermite-Fubini Numbers and Polynomials

2019 ◽  
Author(s):  
Waseem Khan ◽  
Nisar K S ◽  
Idrees A. Khan
Keyword(s):  
Generating Functions ◽  
New Class ◽  
Summation Formulas ◽  
Symmetric Identities

In this paper, we introduce a new class of Laguerre-based Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of Laguerre-based Hermite-Fubini numbers and polynomials by using generating functions.

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.

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