scholarly journals A new class of generalized Apostol–Bernoulli polynomials and some analogues of the Srivastava–Pintér addition theorem

2011 ◽  
Vol 24 (11) ◽  
pp. 1888-1893 ◽  
Author(s):  
R. Tremblay ◽  
S. Gaboury ◽  
B.-J. Fugère
Keyword(s):  
Bernoulli Polynomials ◽  
Addition Theorem ◽  
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 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).

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 Some identities and recurrence relations on the two variables Bernoulli, Euler and Genocchi polynomials

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1757-1765
Author(s):  
Veli Kurt ◽  
Burak Kurt
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Addition Theorem ◽  
Euler Polynomials ◽  
Genocchi Polynomials

Mahmudov in ([16], [17], [18]) introduced and investigated some q-extensions of the q-Bernoulli polynomials B(?)n,q (x,y) of order ?, the q-Euler polynomials ?(?)n,q (x,y) of order ? and the q-Genocchi polynomials G(?)n,q (x,y) of order ?. In this article, we give some identities for the q-Bernoulli polynomials, q-Euler polynomials and q-Genocchi polynomials and the recurrence relation between these polynomials. We give a different form of the analogue of the Srivastava-Pint?r addition theorem.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
R. Tremblay ◽  
S. Gaboury ◽  
B.-J. Fugère
Keyword(s):  
Addition Theorem ◽  
Euler Polynomials ◽  
Addition Formula ◽  
New Class ◽  
Genocchi Polynomials

The main object of this paper is to introduce and investigate two new classes of generalized Apostol-Euler and Apostol-Genocchi polynomials. In particular, we obtain a new addition formula for the new class of the generalized Apostol-Euler polynomials. We also give an extension and some analogues of the Srivastava-Pintér addition theorem obtained in the works by Srivastava and Pintér (2004) and R. Tremblay, S. Gaboury, B.-J. Fugère, and Tremblay et al. (2011). for both classes.

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 -Analogues of the Bernoulli and Genocchi Polynomials and the Srivastava-Pintér Addition Theorems

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
N. I. Mahmudov
Keyword(s):  
Addition Theorem ◽  
Addition Theorems ◽  
New Class ◽  
Genocchi Polynomials

The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli and Genocchi polynomials based on the -integers. The -analogues of well-known formulas are derived. The -analogue of the Srivastava-Pintér addition theorem is obtained.

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.

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.

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