scholarly journals Truncated Fubini Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 431 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Summation Formulas

In this paper, we introduce the two-variable truncated Fubini polynomials and numbers and then investigate many relations and formulas for these polynomials and numbers, including summation formulas, recurrence relations, and the derivative property. We also give some formulas related to the truncated Stirling numbers of the second kind and Apostol-type Stirling numbers of the second kind. Moreover, we derive multifarious correlations associated with the truncated Euler polynomials and truncated Bernoulli polynomials.

scholarly journals Certain Results on (p,q)-Hermite Based Apostol Type Frobenius-Euler Polynomials

2019 ◽  
Author(s):  
Waseem Khan ◽  
Idrees Ahmad Khan ◽  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Difference Equations ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Integral Representations ◽  
Euler Polynomials ◽  
Addition Theorems ◽  
Summation Formulas ◽  
Euler Polynomials And Numbers

In the present paper, the (p,q)-Hermite based Apostol type Frobenius-Euler polynomials and numbers are firstly considered and then diverse basic identities and properties for the mentioned polynomials and numbers, including addition theorems, difference equations, integral representations, derivative properties, recurrence relations. Moreover, we provide summation formulas and relations associated with the Stirling 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 A Note on $(P,Q)$-Analogue Type of Fubini Numbers and Polynomials

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.

scholarly journals Two-Variable Type 2 Poly-Fubini Polynomials

Mathematics ◽  
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.

scholarly journals On Degenerate Truncated Special Polynomials

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 144 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Bernstein Polynomials ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Euler Polynomials ◽  
Exponential Polynomials ◽  
Special Polynomials ◽  
Summation Formulas ◽  
Special Cases ◽  
Proof Techniques

The main aim of this paper is to introduce the degenerate truncated forms of multifarious special polynomials and numbers and is to investigate their various properties and relationships by using the series manipulation method and diverse special proof techniques. The degenerate truncated exponential polynomials are first considered and their several properties are given. Then the degenerate truncated Stirling polynomials of the second kind are defined and their elementary properties and relations are proved. Also, the degenerate truncated forms of the bivariate Fubini and Bell polynomials and numbers are introduced and various relations and formulas for these polynomials and numbers, which cover several summation formulas, addition identities, recurrence relationships, derivative property and correlations with the degenerate truncated Stirling polynomials of the second kind, are acquired. Thereafter, the truncated degenerate Bernoulli and Euler polynomials are considered and multifarious correlations and formulas including summation formulas, derivation rules and correlations with the degenerate truncated Stirling numbers of the second are derived. In addition, regarding applications, by introducing the degenerate truncated forms of the classical Bernstein polynomials, we obtain diverse correlations and formulas. Some interesting surface plots of these polynomials in the special cases are provided.

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 On the generalized q-poly-Euler polynomials of the second kind

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 475-482
Author(s):  
Veli Kurt
Keyword(s):  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Basic Properties

In this work, we define the generalized q-poly-Euler numbers of the second kind of order ? and the generalized q-poly-Euler polynomials of the second kind of order ?. We investigate some basic properties for these polynomials and numbers. In addition, we obtain many identities, relations including the Roger-Sz?go polynomials, the Al-Salam Carlitz polynomials, q-analogue Stirling numbers of the second kind and two variable Bernoulli 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 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.

On applications for Mahler expansion associated with p-adic q-integrals

2019 ◽  
Vol 15 (01) ◽  
pp. 67-84 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Explicit Formula ◽  
Gamma Function ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Symmetric Relation ◽  
Euler Constant ◽  
Basic Properties ◽  
Changhee Polynomials

In this paper, we primarily consider a generalization of the fermionic [Formula: see text]-adic [Formula: see text]-integral on [Formula: see text] including the parameters [Formula: see text] and [Formula: see text] and investigate its some basic properties. By means of the foregoing integral, we introduce two generalizations of [Formula: see text]-Changhee polynomials and numbers as [Formula: see text]-Changhee polynomials and numbers with weight [Formula: see text] and [Formula: see text]-Changhee polynomials and numbers of second kind with weight [Formula: see text]. For the mentioned polynomials, we obtain new and interesting relationships and identities including symmetric relation, recurrence relations and correlations associated with the weighted [Formula: see text]-Euler polynomials, [Formula: see text]-Stirling numbers of the second kind and Stirling numbers of first and second kinds. Then, we discover multifarious relationships among the two types of weighted [Formula: see text]-Changhee polynomials and [Formula: see text]-adic gamma function. Also, we compute the weighted fermionic [Formula: see text]-adic [Formula: see text]-integral of the derivative of [Formula: see text]-adic gamma function. Moreover, we give a novel representation for the [Formula: see text]-adic Euler constant by means of the weighted [Formula: see text]-Changhee polynomials and numbers. We finally provide a quirky explicit formula for [Formula: see text]-adic Euler constant.

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