• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1605-1621
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• 2017

The purpose of this paper is to construct a new family of the special numbers which are related to the Fubini type numbers and the other well-known special numbers such as the Apostol-Bernoulli numbers, the Frobenius-Euler numbers and the Stirling numbers. We investigate some fundamental properties of these numbers and polynomials. By using generating functions and their functional equations, we derive various formulas and relations related to these numbers and polynomials. In order to compute the values of these numbers and polynomials, we give their recurrence relations. We give combinatorial sums including the Fubini type numbers and the others. Moreover, we give remarks and observation on these numbers and polynomials.

• Communications of the Korean Mathematical Society
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• v.35 no.1
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• pp.21-41
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• 2020

In this paper, we establish an explicit formula for r-Dowling numbers in terms of r-Whitney Lah and r-Whitney numbers of the second kind. This is a generalization of the Qi formula for Bell numbers in terms of Lah and Stirling numbers of the second kind. Moreover, we define the q, r-Dowling numbers, q, r-Whitney Lah numbers and q, r-Whitney numbers of the first kind and obtain several fundamental properties of these numbers such as orthogonality and inverse relations, recurrence relations, and generating functions. Hence, we derive an analogous Qi formula for q, r-Dowling numbers expressed in terms of q, r-Whitney Lah numbers and q, r-Whitney numbers of the second kind.

• Journal of applied mathematics & informatics
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• v.40 no.1_2
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• pp.243-257
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• 2022

In this paper, we present and prove an new theorem on symmetric functions. By using this theorem, we derive some new generating functions of the products of (p, q)-Fibonacci numbers, (p, q)-Lucas numbers, (p, q)-Pell numbers, (p, q)-Pell Lucas numbers, (p, q)-Jacobsthal numbers and (p, q)-Jacobsthal Lucas numbers with Chebyshev polynomials of the first kind.

• Journal of the Korean Mathematical Society
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• v.56 no.1
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• pp.265-284
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• 2019

The main purpose of this paper is to investigate the q-Apostol type Frobenius-Euler numbers and polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive q-integers. By using infinite series representation for q-Apostol type Frobenius-Euler numbers and polynomials including their interpolation functions, we not only give some identities and relations for these numbers and polynomials, but also define generating functions for new numbers and polynomials. Further we give remarks and observations on generating functions for these new numbers and polynomials. By using these generating functions, we derive recurrence relations and finite sums related to these numbers and polynomials. Moreover, by applying higher-order derivative to these generating functions, we derive some new formulas including the Hurwitz-Lerch zeta function, the Apostol-Bernoulli numbers and the Apostol-Euler numbers. Finally, for an application of the generating functions, we derive a multiplication formula, which is very important property in the theories of normalized polynomials and Dedekind type sums.

• Journal of applied mathematics & informatics
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• v.32 no.3_4
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• pp.465-474
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• 2014

Many mathematicians have studied various relations beween Euler number $E_n$, Bernoulli number $B_n$ and Genocchi number $G_n$ (see [1-18]). They have found numerous important applications in number theory. Howard, T.Agoh, S.-H.Rim have studied Genocchi numbers, Bernoulli numbers, Euler numbers and polynomials of these numbers [1,5,9,15]. T.Kim, M.Cenkci, C.S.Ryoo, L. Jang have studied the q-extension of Euler and Genocchi numbers and polynomials [6,8,10,11,14,17]. In this paper, our aim is introducing and investigating an extension term of generalized Euler polynomials. We also obtain some identities and relations involving the Euler numbers and the Euler polynomials, the Genocchi numbers and Genocchi polynomials.

• Bulletin of the Korean Mathematical Society
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• v.41 no.2
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• pp.299-306
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• 2004

In this paper, we shall construct generating function of twisted generalized Euler numbers. By using this function, we shall define twisted generalized Euler polynomials and numbers. We shall give some basic properties of these polynomials and numbers.

• Journal for History of Mathematics
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• v.27 no.1
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• pp.67-79
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• 2014

When imaginary numbers were first encountered in the 16th century, mathematicians were able to calculate the imaginary numbers the same as they are today. However, it required 200 years to mathematically acknowledge the existence of imaginary numbers. The new mathematical situation that arose with a development in mathematics required a harmony of real numbers and imaginary numbers. As a result, the concept of complex number became clear. A history behind the development of complex numbers involved a process of determining a comprehensive perspective that ties real numbers and imaginary numbers in a single category, complex numbers. This came after a resolution of conflict between real numbers and imaginary numbers. This study identified the new perspective and way of mathematical thinking emerging from resolving the conflicts. Also educational implications of the analysis were discussed.

• Honam Mathematical Journal
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• v.35 no.3
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• pp.373-379
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• 2013

In contrast with numerous identities involving the binomial coefficients and the Stirling numbers of the first and second kinds, a few identities involving the Eulerian numbers have been known. The objective of this note is to present certain interesting and (presumably) new identities involving the Eulerian numbers by mainly making use of Worpitzky's identity.

• Journal of the Korean Mathematical Society
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• v.40 no.6
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• pp.963-975
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• 2003

The aim of this work is to construct twisted q-L-series which interpolate twisted q-generalized Bernoulli numbers. By using generating function of q-Bernoulli numbers, twisted q-Bernoulli numbers and polynomials are defined. Some properties of this polynomials and numbers are described. The numbers $L_{q}(1-n,\;X,\;{\xi})$ is also given explicitly.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.3
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• pp.591-599
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• 2013

A large number of sequences of polynomials and numbers have arisen in mathematics. Some of them, for example, Bernoulli polynomials and numbers, have been investigated deeply and widely. Here we aim at presenting certain interesting and (potentially) useful identities involving mainly in the second-order Eulerian numbers and Stirling polynomials, which seem to have not been given so much attention.