scholarly journals A Note on a New Type of Degenerate poly-Euler Numbers and Polynomials

2020 ◽  
Author(s):  
Waseem Khan
Keyword(s):  
Arithmetic Function ◽  
Bernoulli Numbers ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Polylogarithm Function ◽  
Euler Numbers And Polynomials ◽  
Special Numbers And Polynomials ◽  
Euler Polynomials And Numbers ◽  
New Type ◽  
Logarithm Function

Kim-Kim [12] introduced the new type of degenerate Bernoulli numbers and polynomials arising from the degenerate logarithm function. In this paper, we introduce a new type of degenerate poly-Euler polynomials and numbers, are called degenerate poly-Euler polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Euler polynomials and numbers. In the last section, we also consider the degenerate unipoly-Euler polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

scholarly journals A ( p , q ) analogue of Poly-Euler polynomials and some related polynomials

2020 ◽  
Vol 72 (4) ◽  
pp. 467-482
Author(s):  
T. Komatsu ◽  
J. L. Ramírez ◽  
V. F. Sirvent
Keyword(s):  
Bernoulli Polynomials ◽  
Combinatorial Identities ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Polylogarithm Function ◽  
Euler Numbers And Polynomials ◽  
Euler Polynomials And Numbers

UDC 517.5 We introduce a ( p , q ) -analogue of the poly-Euler polynomials and numbers by using the ( p , q ) -polylogarithm function.  These new sequences are generalizations of the poly-Euler numbers and polynomials.  We give several combinatorial identities and properties of these new polynomials, and also show some relations with ( p , q ) -poly-Bernoulli polynomials and ( p , q ) -poly-Cauchy polynomials. The ( p , q ) -analogues generalize the well-known concept of the q -analogue.

On (ρ,q)-Euler numbers and polynomials associated with (ρ,q)-Volkenborn integrals

2017 ◽  
Vol 14 (01) ◽  
pp. 241-253 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Bernoulli Numbers ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Type Formula ◽  
Bernoulli Numbers And Polynomials ◽  
Euler Numbers And Polynomials ◽  
Euler Polynomials And Numbers ◽  
Haar Distribution

Recently, Araci et al. introduced [Formula: see text]-analogue of the Haar distribution and by means of the distribution, they constructed [Formula: see text]-Volkenborn integral yielding to Carlitz-type [Formula: see text]-Bernoulli numbers and polynomials. The aim of the present paper is to introduce a generalization of the fermionic [Formula: see text]-adic measure based on [Formula: see text]-integers and set the corresponding integral to this measure. Consequently, Carlitz-type [Formula: see text]-Euler polynomials and numbers are defined in terms of the above mentioned integral. Moreover, some of their identities and properties are established.

scholarly journals Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 47 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Bernstein Polynomials ◽  
Umbral Calculus ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Special Numbers And Polynomials ◽  
Degenerate Bernstein Polynomials ◽  
Combinatorial Methods

In recent years, intensive studies on degenerate versions of various special numbers and polynomials have been done by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations. The degenerate Bernstein polynomials and operators were recently introduced as degenerate versions of the classical Bernstein polynomials and operators. Herein, we firstly derive some of their basic properties. Secondly, we explore some properties of the degenerate Euler numbers and polynomials and also their relations with the degenerate Bernstein polynomials.

scholarly journals Type 2 Degenerate Poly-Euler Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1011 ◽  
Author(s):  
Dae Sik Lee ◽  
Hye Kyung Kim ◽  
Lee-Chae Jang
Keyword(s):  
Final Section ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
Euler Polynomials And Numbers ◽  
New Type ◽  
Explicit Expressions

In recent years, many mathematicians have studied the degenerate versions of many special polynomials and numbers. The polyexponential functions were introduced by Hardy and rediscovered by Kim, as inverses to the polylogarithms functions. The paper is divided two parts. First, we introduce a new type of the type 2 poly-Euler polynomials and numbers constructed from the modified polyexponential function, the so-called type 2 poly-Euler polynomials and numbers. We show various expressions and identities for these polynomials and numbers. Some of them involving the (poly) Euler polynomials and another special numbers and polynomials such as (poly) Bernoulli polynomials, the Stirling numbers of the first kind, the Stirling numbers of the second kind, etc. In final section, we introduce a new type of the type 2 degenerate poly-Euler polynomials and the numbers defined in the previous section. We give explicit expressions and identities involving those polynomials in a similar direction to the previous section.

scholarly journals Viewing Some Ordinary Differential Equations from the Angle of Derivative Polynomials

2016 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Generalized Derivative ◽  
Bernoulli Numbers And Polynomials ◽  
Euler Numbers And Polynomials

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

scholarly journals On A New Type of Degenerate Poly-Fubini Numbers and Polynomials

2020 ◽  
Author(s):  
Waseem Khan
Keyword(s):  
Arithmetic Function ◽  
Polylogarithm Function ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Explicit Expressions

In this paper, we introduce a new type of degenerate poly-Fubini polynomials and numbers, are called degenerate poly-Fubini polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Fubini polynomials and numbers. In the last section, we also consider the degenerate unipoly-Fubini polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

scholarly journals A Note on the -Euler Numbers and Polynomials with Weak Weight

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
H. Y. Lee ◽  
N. S. Jung ◽  
C. S. Ryoo
Keyword(s):  
Numerical Experiments ◽  
Regular Structure ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Complex Roots ◽  
New Type

We construct a new type of -Euler numbers and polynomials with weak weight : , , respectively. Some interesting results and relationships are obtained. Also, we observe the behavior of roots of the -Euler numbers and polynomials with weak weight . By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of -Euler polynomials with weak weight .

scholarly journals NOTE ON q-DEDEKIND-TYPE SUMS RELATED TO q-EULER POLYNOMIALS

2011 ◽  
Vol 54 (1) ◽  
pp. 121-125 ◽  
Author(s):  
TAEKYUN KIM
Keyword(s):  
Continuous Function ◽  
Zeta Function ◽  
Higher Order ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Euler Polynomials And Numbers

AbstractRecently, q-Dedekind-type sums related to q-zeta function and basic L-series are studied by Simsek in [13] (Y. Simsek, q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), 333–351) and Dedekind-type sums related to Euler numbers and polynomials are introduced in the previous paper [11] (T. Kim, Note on Dedekind type DC sums, Adv. Stud. Contem. Math. 18 (2009), 249–260). It is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of the higher order Dedekind the type sums related to q-Euler polynomials and numbers by using an invariant p-adic q-integrals.

scholarly journals A New Family of Degenerate Poly-Genocchi Polynomials with Its Certain Properties

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Waseem A. Khan ◽  
Rifaqat Ali ◽  
Khaled Ahmad Hassan Alzobydi ◽  
Naeem Ahmed
Keyword(s):  
Arithmetic Function ◽  
Polylogarithm Function ◽  
Genocchi Polynomials ◽  
New Family ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Explicit Expressions

In this paper, we introduce a new type of degenerate Genocchi polynomials and numbers, which are called degenerate poly-Genocchi polynomials and numbers, by using the degenerate polylogarithm function, and we derive several properties of these polynomials systematically. Then, we also consider the degenerate unipoly-Genocchi polynomials attached to an arithmetic function, by using the degenerate polylogarithm function, and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and 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).

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