• 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.

• 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 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.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.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.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1463-1481
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• 2021

Divisibility properties of harmonic numbers by a prime number p have been a recurrent topic. However, finding the exact p-adic orders of them is not easy. Using class numbers of number fields and Bernoulli numbers, we compute the exact p-adic orders of harmonic type sums. Moreover, we obtain an asymptotic formula for generalized harmonic numbers whose p-adic orders are exactly one.

• Communications of the Korean Mathematical Society
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• v.12 no.4
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• pp.829-839
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• 1997

We study the Betti numbers, the Bass numbers and the resolution of modules under the change of rings. For modules of finite homological dimension, we study the Euler characteristic of them.

• 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.

• Journal of the Korean Mathematical Society
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• v.43 no.1
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• pp.183-198
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• 2006

In this paper q-extensions of Genocchi numbers are defined and several properties of these numbers are presented. Properties of q-Genocchi numbers and polynomials are used to construct q-extensions of p-adic measures which yield to obtain p-adic interpolation functions for q-Genocchi numbers. As an application, general systems of congruences, including Kummer-type congruences for q-Genocchi numbers are proved.