• 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.

• 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.

• 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.51 no.5
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• pp.1045-1073
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• 2014

The study of the identities of symmetry for the Bernoulli polynomials arises from the study of Gauss's multiplication formula for the gamma function. There are many works in this direction. In the sense of p-adic analysis, the q-Bernoulli polynomials are natural extensions of the Bernoulli and Apostol-Bernoulli polynomials (see the introduction of this paper). By using the N-fold iterated Volkenborn integral, we derive serval identities of symmetry related to the q-extension power sums and the higher order q-Bernoulli polynomials. Many previous results are special cases of the results presented in this paper, including Tuenter's classical results on the symmetry relation between the power sum polynomials and the Bernoulli numbers in [A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001), no. 3, 258-261] and D. S. Kim's eight basic identities of symmetry in three variables related to the q-analogue power sums and the q-Bernoulli polynomials in [Identities of symmetry for q-Bernoulli polynomials, Comput. Math. Appl. 60 (2010), no. 8, 2350-2359].

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1149-1156
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• 2016

In this paper, we introduce the degenerate Carlitz q-Bernoulli numbers and polynomials and give some interesting identities and properties of these numbers and polynomials which are derived from the generating functions and p-adic integral equations.

• Journal of the Korean Mathematical Society
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• v.44 no.4
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• pp.915-929
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• 2007

The main result of this paper is to construct the derivative twisted p-adic (h, q)-L-functions at s = 0. We obtain twisted version of Theorem 4 in [17]. We also obtain twisted (h, q)-extension of Proposition 1 in [3].

• 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.