• Journal of the Korean Mathematical Society
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• v.39 no.2
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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].

• Journal for History of Mathematics
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• v.22 no.4
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• pp.145-160
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• 2009

In the end of 20th century, the concept of p-adic invariant q-integral was introduced by Taekyun Kim. The p-adic invariant q-integral is the extension of Jackson's q-integral on complex space. It is also considered as the answer of the question whether the ultra non-archimedian integral exists or not. In this paper, we investigate the background of historical mathematics for the p-adic invariant q-integral on $Z_p$ and the trend of the research in this field at present.

• Bulletin of the Korean Mathematical Society
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• v.40 no.3
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• pp.509-520
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• 2003

Let p be a prime number, $Q_{p}$ the field of p-adic numbers, K a finite extension of $Q_{p}$, $\bar{K}}$ a fixed algebraic closure of K and $C_{p}$ the completion of K with respect to the p-adic valuation. Let E be a closed subfield of $C_{p}$, containing K. Given elements $t_1$...,$t_{r}$ $\in$ E for which the field K($t_1$...,$t_{r}$) is dense in E, we construct integral bases of E over K.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.2
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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.

• Kyungpook Mathematical Journal
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• v.52 no.3
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• pp.299-306
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• 2012

We will give a new proof of the Lebesgue-Radon-Nikodym theorem with respect to weighted p-adic q-measure on $Z_p$, using Mahler expansion of continuous functions, studied by the authors in 2012. In the special case, q = 1, we can derive the same result as in Kim, 2012, Kim et al, 2011.

• Honam Mathematical Journal
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• v.33 no.4
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• pp.529-534
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• 2011

In this paper, we consider the q-analogues of Euler numbers and polynomials using the fermionic p-adic invariant integral on $\mathbb{Z}_p$. From these numbers and polynomials, we derive some interesting identities and properties on the q-analogues of Euler numbers and polynomials.

• Honam Mathematical Journal
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• v.34 no.1
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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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• v.31 no.3_4
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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.

• Journal of the Korean Mathematical Society
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• v.43 no.1
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• pp.111-131
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• 2006

The goal of this paper is to define p-adic Hardy sums and p-adic q-higher-order Hardy-type sums. By using these sums and p-adic q-higher-order Dedekind sums, we construct p-adic continuous functions for an odd prime. These functions contain padic q-analogue of higher-order Hardy-type sums. By using an invariant p-adic q-integral on $\mathbb{Z}_p$, we give fundamental properties of these sums. We also establish relations between p-adic Hardy sums, Bernoulli functions, trigonometric functions and Lambert series.

• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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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.