• Journal of applied mathematics & informatics
• /
• v.32 no.3_4
• /
• pp.465-474
• /
• 2014

Many mathematicians have studied various relations beween Euler number $E_n$, Bernoulli number $B_n$ and Genocchi number $G_n$ (see [1-18]). They have found numerous important applications in number theory. Howard, T.Agoh, S.-H.Rim have studied Genocchi numbers, Bernoulli numbers, Euler numbers and polynomials of these numbers [1,5,9,15]. T.Kim, M.Cenkci, C.S.Ryoo, L. Jang have studied the q-extension of Euler and Genocchi numbers and polynomials [6,8,10,11,14,17]. In this paper, our aim is introducing and investigating an extension term of generalized Euler polynomials. We also obtain some identities and relations involving the Euler numbers and the Euler polynomials, the Genocchi numbers and Genocchi polynomials.

• Communications of the Korean Mathematical Society
• /
• v.36 no.1
• /
• pp.19-26
• /
• 2021

Degenerate versions of the special polynomials and numbers since they have many applications in analytic number theory, combinatorial analysis and p-adic analysis. In this paper, we define the degenerate poly-Euler numbers and polynomials arising from the modified polyexponential functions. We derive explicit relations for these numbers and polynomials. Also, we obtain some identities involving these polynomials and some other special numbers and polynomials.

• Journal of applied mathematics & informatics
• /
• v.35 no.5_6
• /
• pp.599-609
• /
• 2017

We construct degenerated Euler polynomials of the second kind and find some basic properties of this polynomials. From this paper, we can see degenerated alternative power sum is defined and is related to degenerated Euler polynomials of the second kind. Using this power sum, we have a number of symmetric properties of degenerated Euler polynomials of the second kind.

• Journal of applied mathematics & informatics
• /
• v.29 no.5_6
• /
• pp.1221-1228
• /
• 2011

Recently, several mathematicians have studied some interesting relations between extended q-Euler number and Bernstein polynomials(see [3, 5, 7, 8, 10]). In this paper, we give some interesting identities on the Genocchi polynomials and Bernstein polynomials.

• Kyungpook Mathematical Journal
• /
• v.64 no.1
• /
• pp.133-159
• /
• 2024

In a previous work we studied generalized Stirling numbers of the second kind S^{(a₂,b₂,p₂)}_a1,b1 (p₁, k), where a₁, a₂, b₁, b₂ are given complex numbers, a₁, a₂ ≠ 0, and p₁, p₂ are non-negative integers given. In this work we use these generalized Stirling numbers to define Bernoulli polynomials in two variables B_p1,p2 (x₁, x₂), and Euler polynomials in two variables E_p1p2 (x₁, x₂). By using results for S^{(1,x₂,p₂)}_1,x1 (p₁, k), we obtain generalizations, to the bivariate case, of some well-known properties from the standard case, as addition formulas, difference equations and sums of powers. We obtain some identities for bivariate Bernoulli and Euler polynomials, and some generalizations, to the bivariate case, of several known identities for Bernoulli and Euler numbers and polynomials of the standard case.

• Journal of applied mathematics & informatics
• /
• v.36 no.5_6
• /
• pp.369-379
• /
• 2018

We consider a p-adic Euler L-function of two variables which interpolate the generalized Euler polynomials at nonpositive integers. We also show that the reflection formula and the functional equation for these functions.

• Journal of applied mathematics & informatics
• /
• v.35 no.3_4
• /
• pp.413-421
• /
• 2017

In this paper, we study the generalizations of Euler numbers and polynomials by using the q-extension with p-adic integral on $\mathbb{Z}_p$. We call these: the generalized q-${\omega}$-Euler numbers $E^{({\alpha})}_{n,q,{{\omega}}(a)$ and polynomials $E^{({\alpha})}_{n,q,{\omega}}(x;a)$. We investigate some elementary properties and relations for $E^{({\alpha})}_{n,q,{{\omega}}(a)$ and $E^{({\alpha})}_{n,q,{\omega}}(x;a)$.

• Bulletin of the Korean Mathematical Society
• /
• v.47 no.6
• /
• pp.1181-1188
• /
• 2010

q-Volkenborn integrals ([8]) and fermionic invariant q-integrals ([12]) are introduced by T. Kim. By using these integrals, Euler q-zeta functions are introduced by T. Kim ([18]). Then, by using the Euler q-zeta functions, S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin ([25]) studied q-Genocchi zeta functions. And also Y. H. Kim, W. Kim, and C. S. Ryoo ([7]) investigated twisted q-zeta functions and their applications. In this paper, we consider the q-analogue of twisted Lerch type Euler zeta functions defined by $${\varsigma}E,q,\varepsilon(s)=[2]q \sum\limits_{n=0}^\infty\frac{(-1)^n\epsilon^nq^{sn}}{[n]_q}$$ where 0 < q < 1, $\mathfrak{R}$(s) > 1, $\varepsilon{\in}T_p$, which are compared with Euler q-zeta functions in the reference ([18]). Furthermore, we give the q-extensions of the above twisted Lerch type Euler zeta functions at negative integers which interpolate twisted q-Euler polynomials.

• Bulletin of the Korean Mathematical Society
• /
• v.36 no.1
• /
• pp.47-61
• /
• 1999

In this paper we difine poly-Euler numbers which generalize ordinary Euler numbers. We construct a p-adic poly-Euler measure by the poly-Euler polynomials and derive an integral formula.

• Journal of applied mathematics & informatics
• /
• v.41 no.2
• /
• pp.439-448
• /
• 2023

We introduce several higher-order (p, q)-differential equation of which are related to (p, q)-Bernoulli polynomials. We also find some relations between (p, q)-Bernoulli, (p, q)-Euler, and (p, q)-Genocchi polynomials.