• 한국수학사학회지
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• 제22권4호
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• pp.145-160
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• 2009

20세기말 p-진 공간에서 p-진 q-적분의 개념이 김태균에 의해서 처음 도입 되었다([11]). 이러한 적분은 복소수 공간에서 잭슨의 q-적분을 p-진 공간으로 확장 시킨 것이며 또한 울트라 비 아르키메디언 적분의 존재성에 대한 질문의 답으로 볼 수 있다. 본 논문에서는 이러한 p-진 q-적분의 수학사적 배경을 살펴보고, 현재 어떠한 방향으로 연구가 진행되고 있는지를 고찰한다.

• 대한수학회논문집
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• 제38권3호
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• pp.755-772
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• 2023

Our aim is to establish certain image formulas of the (p, q)-extended modified Bessel function of the second kind M_ν,p,q(z) by employing the Marichev-Saigo-Maeda fractional calculus (integral and differential) operators including their composition formulas and using certain integral transforms involving (p, q)-extended modified Bessel function of the second kind M_ν,p,q(z). Corresponding assertions for the Saigo's, Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential operators are deduced. All the results are represented in terms of the Hadamard product of the (p, q)-extended modified Bessel function of the second kind M_ν,p,q(z) and Fox-Wright function _rΨ_s(z).

• 대한수학회지
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• 제39권2호
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• pp.221-236
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• 2002

Using the p-adic q-integral due to T. Kim[4], we define a number B＊$_{n}$(q) and a polynomial B＊$_{n}$(q) which are p-adic q-analogue of the ordinary Bernoulli number and Bernoulli polynomial, respectively. We investigate some properties of these. Also, we give slightly different construction of Tsumura＇s p-adic function $\ell$$_{p}$(u, s, $\chi$) [14] using the p-adic q-integral in [4].n [4].

• 충청수학회지
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• 제23권2호
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• pp.207-213
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• 2010

In this paper, we define the q-extension of the Hardy-Littlewood-type maximal operator related to q-Volkenborn integral. By the meaning of the extension of q-Volkenborn integral, we obtain the boundedness of the q-extension of the Hardy-Littlewood-type maximal operator in the p-adic integer ring.

• Journal of applied mathematics & informatics
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• 제37권3_4호
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• pp.295-305
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• 2019

In the present paper, we introduce (p, q)-Frobenius-Genocchi numbers and polynomials and investigate some basic identities and properties for these polynomials and numbers including addition theorems, difference equations, derivative properties, recurrence relations and so on. Then, we provide integral representations, implicit and explicit formulas and relations for these polynomials and numbers. We consider some relationships for (p, q)-Frobenius-Genocchi polynomials of order ${\alpha}$ associated with (p, q)-Bernoulli polynomials, (p, q)-Euler polynomials and (p, q)-Genocchi polynomials.

• Journal of applied mathematics & informatics
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• 제41권1호
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• pp.167-179
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• 2023

In this paper, we investigate solutions of equations which are linear (p, q)-difference equation of higher order by using (p, q)-derivative and integral. We also derive the solution of equation in the case of second order.

• 대한수학회보
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• 제44권1호
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• pp.1-12
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• 2007

In this paper, we consider the q-Volkenborn integral of uniformly differentiable functions on the p-adic integer ring. By using this integral, we obtain the generating functions of twisted q-generalized Bernoulli numbers and polynomials. We find some properties of these numbers and polynomials.

• 대한수학회지
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• 제55권3호
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• pp.671-694
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• 2018

In this paper, we investigate the weighted vector-valued bounds for a class of multilinear singular integral operators, and its commutators, from $L^{p_1}(l^{q_1};\;{\mathbb{R}}^n,\;w_1){\times}{\cdots}{\times}L^{p_m}(l^{q_m};\;{\mathbb{R}}^n,\;w_m)$ to $L^p(l^q;\;{\mathbb{R}}^n,\;{\nu}_{\vec{w}})$, with $p_1,{\cdots},p_m$, $q_1,{\cdots},q_m{\in}(1,\;{\infty})$, $1/p=1/p_1+{\cdots}+1/p_m$, $1/q=1/q_1+{\cdots}+1/q_m$ and ${\vec{w}}=(w_1,{\cdots},w_m)$ a multiple $A_{\vec{P}}$ weights. Our argument also leads to the weighted weak type endpoint estimates for the commutators. As applications, we obtain some new weighted estimates for the $Calder{\acute{o}}n$ commutator.

• 호남수학학술지
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• 제33권4호
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• pp.519-527
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• 2011

Recently, the modified q-Bernoulli numbers and polynomials with weight ${\alpha}$ are introduced in [3]: In this paper we give some interesting p-adic integral representation on $\mathbb{Z}_p$ of the weighted q-Bernstein polynomials related to the modified q-Bernoulli numbers and polynomials with weight ${\alpha}$. From those integral representation on $\mathbb{Z}_p$ of the weighted q-Bernstein polynomials, we can derive some identities on the modified q-Bernoulli numbers and polynomials with weight ${\alpha}$.

• Journal of applied mathematics & informatics
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• 제31권3_4호
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• pp.365-372
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• 2013

The essential aim of this paper is to define weighted $q$-Hardylittlewood-type maximal operator by means of $p$-adic $q$-invariant distribution on $\mathbb{Z}_p$. Moreover, we give some interesting properties concerning this type maximal operator.