• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1427-1437
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• 2016

Cauchy polynomials are also called Bernoulli polynomials of the second kind and these polynomials are very important to study mathematical physics. D. S. Kim et al. have studied some properties of Bernoulli polynomials of the second kind associated with special polynomials arising from umbral calculus. T. Kim introduced the degenerate Cauchy numbers and polynomials which are derived from the degenerate function $e^t$. Recently J. Jeong, S. H. Rim and B. M. Kim studied on finite times degenerate Cauchy numbers and polynomials. In this paper we consider finite times degenerate higher-order Cauchy numbers and polynomials, and give some identities and properties of these polynomials.

• Journal for History of Mathematics
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• v.20 no.4
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• pp.71-84
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• 2007

J. Bernoulli first discovered the method which one can produce those formulae for the sum $S_n(k)=\sum_{{\iota}=1}^n\;{\iota}^k$ for any natural numbers k. After then, there has been increasing interest in Bernoulli and Euler numbers associated with Riemann zeta functions. Recently, Kim have been studied extended q-Bernoulli numbers and q-Euler numbers associated with p-adic q-integral on $\mathbb{Z}_p$, and sums of powers of consecutive q-integers, etc. In this paper, we investigate for the historical background and evolution process of the sums of powers of consecutive q-integers and discuss for Euler zeta functions subjects which are studying related to these areas in the recent.

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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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 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].

• Journal of applied mathematics & informatics
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• v.39 no.5_6
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• pp.643-654
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• 2021

In this paper, we define a modified q-poly-Bernoulli polynomials of the first type and modified q-poly-tangent 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 the Korean Mathematical Society
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• v.44 no.5
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• pp.1163-1184
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• 2007

In this presentation dedicated to the tricentennial birth anniversary of the great eighteenth-century Swiss mathematician, Leonhard Euler (1707-1783), we begin by remarking about the so-called Basler problem of evaluating the Zeta function ${\zeta}(s)$ [in the much later notation of Georg Friedrich Bernhard Riemann (1826-1866)] when s=2, which was then of vital importance to Euler and to many other contemporary mathematicians including especially the Bernoulli brothers [Jakob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748)], and for which a fascinatingly large number of seemingly independent solutions have appeared in the mathematical literature ever since Euler first solved this problem in the year 1736. We then investigate various recent developments on the evaluations and representations of ${\zeta}(s)$ when $s{\in}{\mathbb{N}}{\backslash}\;[1],\;{\mathbb{N}}$ being the set of natural numbers. We emphasize upon several interesting classes of rapidly convergent series representations for ${\zeta}(2n+1)(n{\in}{\mathbb{N}})$ which have been developed in recent years. In two of many computationally useful special cases considered here, it is observed that ${\zeta}(3)$ can be represented by means of series which converge much more rapidly than that in Euler's celebrated formula as well as the series used recently by Roger $Ap\'{e}ry$ (1916-1994) in his proof of the irrationality of ${\zeta}(3)$. Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of one of these series are capable of producing an accuracy of seven decimal places.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.49-58
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• 2005

In this paper we observe the structure of the roots of q-Bernoulli polynomials, ${\beta}_n(w,h{\mid}q)$, using numerical investigation. By numerical experiments, we demonstrate a remarkably regular structure of the real roots of ${\beta}_n(w,h{\mid}q)$ for $-{\frac{1}{5}},-{\frac{1}{2}}$. Finally, we give a table for numbers of real and complex zeros of ${\beta}_n(w,h{\mid}q)$.

• Journal of applied mathematics & informatics
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• v.39 no.1_2
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• pp.45-56
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• 2021

D.S. Kim et al. [9] considered some identities and relations for Korobov type numbers and polynomials. In this work, we investigate the degenerate Korobov type Changhee polynomials and the (p,q)-poly-Korobov polynomials. We give a generalization of the Korobov type Changhee polynomials and the (p,q) poly-Korobov polynomials. We prove some properties and identities and explicit relations for these polynomials.

• Journal of applied mathematics & informatics
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• v.40 no.1_2
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• pp.371-391
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• 2022

In this paper we give some prperties of the poly-cosine tangent polynomials and poly-sine tangent polynomials.

• Journal of applied mathematics & informatics
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• v.38 no.3_4
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• pp.291-300
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• 2020

This type of polynomial is a generating function that substitutes e^λt for e^t in the denominator of the generating function for the Bernoulli polynomial, but polynomials by using this generating function has interesting properties involving the location of the roots. We define these generation functions and observe the properties of the generation functions.