scholarly journals On (q, r, w)-Stirling Numbers of the Second Kind

2017 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Exponential Generating Function

In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de…ned (q, r, w)-Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q, r, w)-Stirling numbers of the second kind and q-Bernoulli polynomials in w.

scholarly journals New type of degenerate Daehee polynomials of the second kind

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sunil Kumar Sharma ◽  
Waseem A. Khan ◽  
Serkan Araci ◽  
Sameh S. Ahmed
Keyword(s):  
Generating Function ◽  
Special Class ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Bernoulli Numbers And Polynomials ◽  
New Type ◽  
Order Degenerate ◽  
Math Phys

Abstract Recently, Kim and Kim (Russ. J. Math. Phys. 27(2):227–235, 2020) have studied new type degenerate Bernoulli numbers and polynomials by making use of degenerate logarithm. Motivated by (Kim and Kim in Russ. J. Math. Phys. 27(2):227–235, 2020), we consider a special class of polynomials, which we call a new type of degenerate Daehee numbers and polynomials of the second kind. By using their generating function, we derive some new relations including the degenerate Stirling numbers of the first and second kinds. Moreover, we introduce a new type of higher-order degenerate Daehee polynomials of the second kind. We also derive some new identities and properties of this type of polynomials.

scholarly journals On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 451 ◽  
Author(s):  
Dae Kim ◽  
Taekyun Kim ◽  
Cheon Ryoo ◽  
Yonghong Yao
Keyword(s):  
Integral Representation ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Finite Product ◽  
Bernoulli Numbers And Polynomials ◽  
Special Values ◽  
Invariant Integrals ◽  
Evaluation Problem ◽  
Double Integrals

The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on Z p . We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on Z p of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found.

scholarly journals Construction of the Type 2 Poly-Frobenius-Genocchi Polynomials with Their Certain Applications

2020 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Genocchi Polynomials ◽  
Definition Of ◽  
Special Case

Motivated by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim, in the present paper, we consider a class of new generating function for the Frobenius-Genocchi polynomials, called the type 2 poly-Frobenius-Genocchi polynomials, by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Frobenius-Genocchi polynomias equal a linear combination of the classical Frobenius-Genocchi polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Frobenius-Genocchi polynomials and Bernoulli polynomials of order k. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim, we introduce the unipoly-Frobenius-Genocchi polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Frobenius-Genocchi polynomials and the classical Frobenius-Genocchi polynomials.

scholarly journals Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Lee-Chae Jang ◽  
Hyunseok Lee ◽  
Hanyoung Kim
Keyword(s):  
Hypergeometric Function ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Gauss Hypergeometric Function ◽  
Eulerian Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
New Family ◽  
Degenerate Type ◽  
Degenerate Version

AbstractA new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function. Motivated by that paper and in the light of the recent interests in finding degenerate versions, we construct the generalized degenerate Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. In addition, we construct the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind.

scholarly journals A Novel Kind of the Type 2 Poly-Fubini Polynomials and Numbers

2020 ◽  
Author(s):  
Waseem Khan
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
New Class ◽  
Definition Of ◽  
Special Case

Motivation by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim [9], in the present paper, we consider a new class of new generating function for the Fubini polynomials, called the type 2 poly-Fubini polynomials by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Fubini polynomials equal a linear combination of the classical of the Fubini polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Fubini polynomials and Bernoulli polynomials of order r. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim [9]. We introduce the type 2 unipoly-Fubini polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Fubini polynomials and the classical Fubini polynomials.

scholarly journals The 2-adic analysis of Stirling numbers of the second kind via higher order Bernoulli numbers and polynomials

2020 ◽  
pp. 1-16
Author(s):  
Arnold Adelberg
Keyword(s):  
Previous Analysis ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Bernoulli Numbers And Polynomials

Several new estimates for the [Formula: see text]-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as to new theorems, including a long-standing open conjecture by Lengyel. The estimates and criteria all depend on our previous analysis of powers of [Formula: see text] in the denominators of coefficients of higher order Bernoulli polynomials. The corresponding estimates for Stirling numbers of the first kind are also proved. Some attention is given to asymptotic cases, which will be further explored in subsequent publications.

scholarly journals An Extended Generalized q-Extensions for the Apostol Type Polynomials

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Letelier Castilla ◽  
William Ramírez ◽  
Alejandro Urieles
Keyword(s):  
Generating Function ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Genocchi Polynomials ◽  
Differential Properties

Through a modification on the parameters associated with generating function of the q-extensions for the Apostol type polynomials of order α and level m, we obtain some new results related to a unified presentation of the q-analog of the generalized Apostol type polynomials of order α and level m. In addition, we introduce some algebraic and differential properties for the q-analog of the generalized Apostol type polynomials of order α and level m and the relation of these with the q-Stirling numbers of the second kind, the generalized q-Bernoulli polynomials of level m, the generalized q-Apostol type Bernoulli polynomials, the generalized q-Apostol type Euler polynomials, the generalized q-Apostol type Genocchi polynomials of order α and level m, and the q-Bernstein polynomials.

scholarly journals On generalized q-poly-Bernoulli numbers and polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 515-520
Author(s):  
Secil Bilgic ◽  
Veli Kurt
Keyword(s):  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Euler Numbers And Polynomials

Many mathematicians in ([1],[2],[5],[14],[20]) introduced and investigated the generalized q-Bernoulli numbers and polynomials and the generalized q-Euler numbers and polynomials and the generalized q-Gennochi numbers and polynomials. Mahmudov ([15],[16]) considered and investigated the q-Bernoulli polynomials B(?)n,q(x,y) in x,y of order ? and the q-Euler polynomials E(?) n,q (x,y)in x,y of order ?. In this work, we define generalized q-poly-Bernoulli polynomials B[k,?] n,q (x,y) in x,y of order ?. We give new relations between the generalized q-poly-Bernoulli polynomials B[k,?] n,q (x,y) in x,y of order ? and the generalized q-poly-Euler polynomials ?[k,?] n,q (x,y) in x,y of order ? and the q-Stirling numbers of the second kind S2,q(n,k).

scholarly journals IDENTITIES OF SYMMETRY FOR GENERALIZED TWISTED BERNOULLI POLYNOMIALS TWISTED BY RAMIFIED ROOTS OF UNITY

2014 ◽  
Vol 60 (1) ◽  
pp. 19-36
Author(s):  
Dae San Kim
Keyword(s):  
Generating Function ◽  
Bernoulli Polynomials ◽  
Roots Of Unity ◽  
Integral Expression ◽  
Power Sums ◽  
Exponential Generating Function

Abstract We derive eight identities of symmetry in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by ramified roots of unity. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the p-adic integral expression of the generating function for the generalized twisted Bernoulli polynomials and the quotient of p-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums.

scholarly journals Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim ◽  
Byungje Lee
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Invariant Integral ◽  
Generalized Bernoulli Polynomials ◽  
Symmetric Properties

By the properties ofp-adic invariant integral onℤp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties ofp-adic invariant integral onℤp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

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