scholarly journals Ordinary and degenerate Euler numbers and polynomials

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim ◽  
Jongkyum Kwon
Keyword(s):  
Stirling Numbers ◽  
Recursive Formula ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Integer Power ◽  
Power Sums ◽  
Power Sum ◽  
Euler Numbers And Polynomials

Abstract In this paper, we study some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials which are derived from the fermionic p-adic integrals on $\mathbb{Z}_{p}$ Z p . Specifically, we obtain a recursive formula for alternating integer power sums and representations of alternating integer power sum polynomials in terms of Euler polynomials and Stirling numbers of the second kind, as well as various properties about Euler numbers and polynomials. In addition, we deduce representations of degenerate alternating integer power sum polynomials in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second kind, as well as certain properties on degenerate Euler numbers and polynomials.

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 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 Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 847 ◽  
Author(s):  
Dmitry V. Dolgy ◽  
Dae San Kim ◽  
Jongkyum Kwon ◽  
Taekyun Kim
Keyword(s):  
Random Variables ◽  
Infinite Family ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Integer Power ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Power Sum ◽  
Invariant Integrals

In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials.

scholarly journals Generalized Tepper’s Identity and Its Application

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 243
Author(s):  
Dmitry Kruchinin ◽  
Vladimir Kruchinin ◽  
Yilmaz Simsek
Keyword(s):  
Number Theory ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Combinatorial Analysis ◽  
Euler Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Euler Numbers And Polynomials

The aim of this paper is to study the Tepper identity, which is very important in number theory and combinatorial analysis. Using generating functions and compositions of generating functions, we derive many identities and relations associated with the Bernoulli numbers and polynomials, the Euler numbers and polynomials, and the Stirling numbers. Moreover, we give applications related to the Tepper identity and these numbers and polynomials.

scholarly journals Identities and relations for Fubini type numbers and polynomials via generating functions and p-adic integral approach

2019 ◽  
Vol 106 (120) ◽  
pp. 113-123
Author(s):  
Neslihan Kilar ◽  
Yilmaz Simsek
Keyword(s):  
Functional Equation ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Combinatorial Analysis ◽  
Integral Approach ◽  
Euler Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Array Polynomials ◽  
Euler Numbers And Polynomials

The Fubini type polynomials have many application not only especially in combinatorial analysis, but also other branches of mathematics, in engineering and related areas. Therefore, by using the p-adic integrals method and functional equation of the generating functions for Fubini type polynomials and numbers, we derive various different new identities, relations and formulas including well-known numbers and polynomials such as the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers of the second kind, the ?-array polynomials and the Lah numbers.

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 Generalized -Euler Numbers and Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
H. Y. Lee ◽  
N. S. Jung ◽  
C. S. Ryoo
Keyword(s):  
Complex Plane ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Interesting Phenomenon ◽  
Euler Numbers And Polynomials

We generalize the Euler numbers and polynomials by the generalized -Euler numbers and polynomials . For the complement theorem, have interesting different properties from the Euler polynomials and we observe an interesting phenomenon of “scattering” of the zeros of the the generalized Euler polynomials in complex plane.

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 Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 9 ◽  
Author(s):  
Daeyeoul Kim ◽  
Yilmaz Simsek ◽  
Ji Suk So
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Integral Formulas ◽  
Euler Numbers And Polynomials ◽  
Negative Order ◽  
Changhee Polynomials ◽  
Changhee Numbers

The purpose of this paper is to construct generating functions for negative order Changhee numbers and polynomials. Using these generating functions with their functional equation, we prove computation formulas for combinatorial numbers and polynomials. These formulas include Euler numbers and polynomials of higher order, Stirling numbers, and negative order Changhee numbers and polynomials. We also give some properties of these numbers and polynomials with their generating functions. Moreover, we give relations among Changhee numbers and polynomials of negative order, combinatorial numbers and polynomials and Bernoulli numbers of the second kind. Finally, we give a partial derivative of an equation for generating functions for Changhee numbers and polynomials of negative order. Using these differential equations, we derive recurrence relations, differential and integral formulas for these numbers and polynomials. We also give p-adic integrals representations for negative order Changhee polynomials including Changhee numbers and Deahee numbers.

scholarly journals Euler Numbers and Polynomials Associated with Zeta Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-11 ◽  
Author(s):  
Taekyun Kim
Keyword(s):  
Zeta Function ◽  
Entire Functions ◽  
Zeta Functions ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials

Fors∈ℂ, the Euler zeta function and the Hurwitz-type Euler zeta function are defined byζE(s)=2∑n=1∞((−1)n/ns), andζE(s,x)=2∑n=0∞((−1)n/(n+x)s). Thus, we note that the Euler zeta functions are entire functions in whole complexs-plane, and these zeta functions have the values of the Euler numbers or the Euler polynomials at negative integers. That is,ζE(−k)=Ek∗, andζE(−k,x)=Ek∗(x). We give some interesting identities between the Euler numbers and the zeta functions. Finally, we will give the new values of the Euler zeta function at positive even integers.

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