• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.125-132
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• 2006

It is the aim of this paper to introduce the Genocchi numbers Gn and polynomials Gn(x) and to display the shape of Genocchi polynomials Gn(x). Finally, we investigate the roots of the Genocchi polynomials Gn(x).

• Journal of Applied and Pure Mathematics
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• 제6권1_2호
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• pp.1-11
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• 2024

In this paper, we define a new fully modified q-poly-Euler numbers and polynomials of the first type by using q-polylogarithm function. We derive some identities of the modified polynomials with Gaussian binomial coefficients. We also explore several relations that are connected with the q-analogue of Stirling numbers of the second kind.

• Journal of applied mathematics & informatics
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• 제37권3_4호
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• pp.219-234
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• 2019

We use the definition of Euler polynomials of the second kind with (p, q)-numbers to identify some identities and properties of these polynomials. We also investigate some relationships between (p, q)-Euler polynomials of the second kind, (p, q)-Bernoulli polynomials, and (p, q)-tangent polynomials by using the properties of (p, q)-exponential function.

• Journal of applied mathematics & informatics
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• 제41권3호
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• pp.591-601
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• 2023

In this note, we give a proof of the p-adic analogue of mild generalization of classical zeta functions by modifying Osipov's method. In addition, we obtain some identities for the p-adic integration, from which, some classical formulas for Euler numbers and polynomials have been deduced.

• Journal of applied mathematics & informatics
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• 제40권5_6호
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• pp.1003-1019
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• 2022

In this paper, by using the multiple fermionic p-adic integrals, we obtain Kummer-type congruences for the higher order Euler numbers and polynomials.

• Journal of applied mathematics & informatics
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• 제33권3_4호
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• pp.305-315
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• 2015

Recently, Jianxin Liu, Hao Pan and Yong Zhang in [On the integral of the product of the Appell polynomials, Integral Transforms Spec. Funct. 25 (2014), no. 9, 680-685] established an explicit formula for the integral of the product of several Appell polynomials. Their work generalizes all the known results by previous authors on the integral of the product of Bernoulli and Euler polynomials. In this note, by using a special case of their formula for Euler polynomials, we shall provide several reciprocity relations between the special values of Tornheim's multiple series.

• Kyungpook Mathematical Journal
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• 제54권3호
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• pp.463-469
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• 2014

In this paper, we construct new q-extension of Euler polynomials with weight 0. These modified q-Euler polynomials are useful to study various identities of Carlitz's q-Bernoulli numbers.

• 호남수학학술지
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• 제34권1호
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• pp.1-9
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• 2012

The purpose of this paper is to give a new construction of the extended q-Euler numbers and polynomials of higher-order with weight by using p-adic q-integral on $\mathbb{Z}_p$.

• Journal of applied mathematics & informatics
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• 제27권3_4호
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• pp.453-462
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• 2009

Kim [4] defined the generalized Genocchi numbers $G_{n,x}$. In this paper, we introduce the generalized Genocchi polynomials $G_{n,x}(x)$. One purpose of this paper is to investigate the zeros of the generalized Genocchi polynomials $G_{n,x}(x)$. We also display the shape of generalized Genocchi polynomials $G_{n,x}(x)$.

• Journal of applied mathematics & informatics
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• 제35권3_4호
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• pp.413-421
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• 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 $E^{({\alpha})}_{n,q,{\omega}}(x;a)$.