Generalized Hermite- based Apostol- Bernoulli, Euler, Genocchi polynomials and their relations

2020 ◽  
Vol 87 (1-2) ◽  
pp. 9
Author(s):  
Aparna Chaturvedi ◽  
Prakriti Rai
Keyword(s):  
Generating Functions ◽  
Euler Polynomials ◽  
Genocchi Polynomials ◽  
Intimate Connection ◽  
Summation Formulae

In this paper, we have generalized Apostol-Hermite-Bernoullli polynomials, Apostol-Hermite-Euler polynomials and Apostol-Hermite-Genocchi polynomials. We have shown that there is an intimate connection between these polynomials and derived some implicit summation formulae by applying the generating functions.

scholarly journals Some New Classes of Generalized Hermite-Based Apostol-Euler and Apostol-Genocchi Polynomials

2015 ◽  
Vol 55 (1) ◽  
pp. 153-170 ◽  
Author(s):  
M. A. Pathan ◽  
Waseem A. Khan
Keyword(s):  
Generating Functions ◽  
Euler Polynomials ◽  
New Class ◽  
Genocchi Polynomials ◽  
Summation Formulae

Abstract In this paper, we introduce a new class of generalized Apostol-Hermite-Euler polynomials and Apostol-Hermite-Genocchi polynomials and derive some implicit summation formulae by applying the generating functions. These results extend some known summations and identities of generalized Hermite-Euler polynomials studied by Dattoli et al, Kurt and Pathan.

scholarly journals Some New Classes of Generalized Lagrange-Based Apostol Type Hermite Polynomials

2018 ◽  
Author(s):  
Waseem A. Khan ◽  
K.S. Nisar
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Hermite Polynomials ◽  
Euler Polynomials ◽  
General Symmetry ◽  
Basic Properties ◽  
Genocchi Polynomials ◽  
General Family ◽  
Summation Formulae ◽  
Explicit Representations

In this paper, we introduce a general family of Lagrange-based Apostol-type Hermite polynomials thereby unifying the Lagrange-based Apostol Hermite-Bernoulli and the Lagrange-based Apostol Hermite-Genocchi polynomials. We also define Lagrange-based Apostol Hermite-Euler polynomials via the generating function. In terms of these generalizations, we find new and useful relations between the unified family and the Apostol Hermite-Euler polynomials. We also derive their explicit representations and list some basic properties of each of them. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions.

scholarly journals A New Class of Laguerre-based Generalized Apostol Polynomials

2016 ◽  
Vol 57 (1) ◽  
pp. 67-89 ◽  
Author(s):  
N.U. Khan ◽  
T. Usman
Keyword(s):  
Generating Functions ◽  
General Symmetry ◽  
Genocchi Numbers ◽  
New Class ◽  
Genocchi Polynomials ◽  
Summation Formulae

Abstract In this paper, we introduce a unified family of Laguerre-based Apostol Bernoulli, Euler and Genocchi polynomials and derive some implicit summation formulae and general symmetry identities arising from different analytical means and applying generating functions. The result extend some known summations and identities of generalized Bernoulli, Euler and Genocchi numbers and polynomials.

scholarly journals A Parametric Kind of Fubini Polynomials of a Complex Variable

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 643
Author(s):  
Sunil Kumar Sharma ◽  
Waseem A. Khan ◽  
Cheon Seoung Ryoo
Keyword(s):  
Complex Variable ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Genocchi Polynomials ◽  
Summation Formulae ◽  
Analytical Properties

In this paper, we propose a parametric kind of Fubini polynomials by defining the two specific generating functions. We also investigate some analytical properties (for example, summation formulae, differential formulae and relationships with other well-known polynomials and numbers) for our introduced polynomials in a systematic way. Furthermore, we consider some relationships for parametric kind of Fubini polynomials associated with Bernoulli, Euler, and Genocchi polynomials and Stirling numbers of the second kind.

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 A New Class of Degenerate Hermite Poly-Genocchi Polynomials

2018 ◽  
Author(s):  
Waseem A. Khan ◽  
K.S. Nisar
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
General Symmetry ◽  
Bernoulli Numbers And Polynomials ◽  
New Class ◽  
Genocchi Polynomials ◽  
Summation Formulae

In this article, authors introduce a new class of degenerate Hermite poly-Genocchi polynomials and give some identities of these polynomials related to the Stirling numbers of the second kind. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of degenerate Hermite poly-Bernoulli numbers and polynomials studied by Khan [8].

scholarly journals Notes on unified q-Apostol-type polynomials

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 921-927 ◽  
Author(s):  
Burak Kurt
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Euler Polynomials ◽  
Addition Theorems ◽  
Genocchi Polynomials

Recently, many mathematicians (Karande and Thakare [6], Ozarslan [14], Ozden et. al. [15], El-Deouky et. al. [5]) have studied the unification of Bernoulli, Euler and Genocchi polynomials. They gave some recurrence relations and proved some theorems. Mahmudov [13] defined the new q-Apostol- Bernoulli and q-Apostol-Euler polynomials. Also he gave the analogous of the Srivastava-Pint?r addition theorems. Kurt [8] gave the new identities and some relations for these polynomials. In this work, we give some recurrence relations for the unified q-Apostol-type polynomials related to multiple sums. By using generating functions we derive many new identities and recurrence relations associated with the q-Apostol type Bernoulli, the q-Apostol-type Euler and the q-Apostol-type Genocchi polynomials and numbers and also the generalized Stirling type numbers of the second kind.

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.

scholarly journals On the three families of extended Laguerre-based Apostol-type polynomials

2021 ◽  
Vol 40 (2) ◽  
pp. 313-334
Author(s):  
M. A. Pathan ◽  
Waseem A. Khan
Keyword(s):  
Generating Functions ◽  
Special Functions ◽  
Hermite Polynomials ◽  
New Class ◽  
Genocchi Polynomials ◽  
Summation Formulae ◽  
Symmetric Identities

In this paper, we introduce a new class of generalized extended Laguerre-based Apostol-type-Bernoulli, Apostol-type-Euler and Apostoltype-Genocchi polynomials. These Apostol type polynomials are used to connect Fubini-Hermite and Bell-Hermite polynomials and to find new representations. We derive some implicit summation formulae and symmetric identities for these families of special functions by applying the generating functions.

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