scholarly journals A new class of generalized polynomials associated with Hermite and poly-Bernoulli polynomials

2021 ◽  
Vol 22 (1) ◽  
pp. 317
Author(s):  
M. A. Pathan ◽  
Waseem A. Khan
Keyword(s):  
Bernoulli Polynomials ◽  
Generalized Polynomials ◽  
New Class

scholarly journals A new class of generalized polynomials associated with Laguerre and Bernoulli polynomials

2019 ◽  
Vol 43 (1) ◽  
pp. 486-497 ◽  
Author(s):  
Nabiullah KHAN ◽  
Talha USMAN ◽  
Junesang CHOI
Keyword(s):  
Bernoulli Polynomials ◽  
Generalized Polynomials ◽  
New Class

scholarly journals A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 648
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran ◽  
Deena Al-Kadi
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Laguerre Polynomials ◽  
Hermite Polynomials ◽  
Bernoulli Polynomials ◽  
Higher Order ◽  
New Class

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

scholarly journals Multifarious Results for q-Hermite Based Frobenius Type Eulerian Polynomials

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.

scholarly journals A New Class of Symmetric Beta Type Distributions Constructed by Means of Symmetric Bernstein Type Basis Functions

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 779 ◽  
Author(s):  
Fusun Yalcin ◽  
Yilmaz Simsek
Keyword(s):  
Bernoulli Polynomials ◽  
Random Variable ◽  
Stirling Numbers ◽  
Moment Generating Function ◽  
Basis Functions ◽  
Integral Representations ◽  
Harmonic Numbers ◽  
Bernstein Type ◽  
Digamma Function ◽  
New Class

The main aim of this paper is to define and investigate a new class of symmetric beta type distributions with the help of the symmetric Bernstein-type basis functions. We give symmetry property of these distributions and the Bernstein-type basis functions. Using the Bernstein-type basis functions and binomial series, we give some series and integral representations including moment generating function for these distributions. Using generating functions and their functional equations, we also give many new identities related to the moments, the polygamma function, the digamma function, the harmonic numbers, the Stirling numbers, generalized harmonic numbers, the Lah numbers, the Bernstein-type basis functions, the array polynomials, and the Apostol–Bernoulli polynomials. Moreover, some numerical values of the expected values for the logarithm of random variable are given.

scholarly journals A Unification of the Generalized Multiparameter Apostol-type Bernoulli, Euler, Fubini, and Genocchi Polynomials of Higher Order

2020 ◽  
Vol 13 (3) ◽  
pp. 587-607
Author(s):  
Nestor Gonzales Acala
Keyword(s):  
Hypergeometric Function ◽  
Explicit Formula ◽  
Higher Order ◽  
Gaussian Hypergeometric Function ◽  
Generalized Polynomials ◽  
New Class ◽  
Genocchi Polynomials

Most unifications of the classical or generalized Bernoulli, Euler, and Genocchi polynomials involve unifying any two or all of the three special types of polynomials (see, [1, 4, 9, 18, 19,21, 24–26, 30, 31]). In this paper, we introduce a new class of multiparameter Fubini-type gener-alized polynomials that unifies four families of higher order generalized Apostol-type polynomials such as the Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi, and Apostol-Fubini polynomials. Moreover, we obtain an explicit formula of these unified generalized polynomials in terms of the Gaussian hypergeometric function, and establish several symmetry identities.

scholarly journals Some (p, q)-analogues of Apostol type numbers and polynomials

2019 ◽  
Vol 23 (1) ◽  
pp. 37-50 ◽  
Author(s):  
Mehmet Acikgoz ◽  
Serkan Araci ◽  
Ugur Duran
Keyword(s):  
Generating Functions ◽  
Bernoulli Polynomials ◽  
Euler Polynomials ◽  
New Class

We consider a new class of generating functions of the generalizations of Bernoulli and Euler polynomials in terms of (p, q)-integers. By making use of these generating functions, we derive (p, q)-generalizations of several old and new identities concerning Apostol–Bernoulli and Apostol–Euler polynomials. Finally, we define the (p, q)-generalization of Stirling polynomials of the second kind of order v, and provide a link between the (p, q)-generalization of Bernoulli polynomials of order v and the (p, q)-generalization of Stirling polynomials of the second kind of order v.

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