• Journal of applied mathematics & informatics
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• v.39 no.1_2
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• pp.57-72
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• 2021

In this paper, we construct q-cosine and sine Genocchi polynomials using q-analogues of addition, subtraction, and q-trigonometric function. From these polynomials, we obtain some properties and identities. We investigate some symmetric properties of q-cosine and sine Genocchi polynomials. Moreover, we find relations between these polynomials and others polynomials.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1289-1296
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• 2013

Recently, T. Kim has introduced and analysed the $q$-Euler polynomials (see [3, 14, 35, 37]). By the same motivation, we will consider some interesting properties of the $q$-Genocchi polynomials. Further, we give some formulae on the Bernstein and $q$-Genocchi polynomials by using $p$-adic integral on $\mathbb{Z}_p$. From these relationships, we establish some interesting identities.

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

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

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

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