• Journal of applied mathematics & informatics
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• v.35 no.5_6
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• pp.449-457
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• 2017

In this paper, we shall provide several properties of Dedekind sums with quasi-periodicity Euler functions. In particular, we present a representation of these Dedekind sums in terms of the Eulerian functions and the tangent functions.

• Journal of applied mathematics & informatics
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• v.39 no.3_4
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• pp.455-467
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• 2021

In this paper, we define twisted q-Euler polynomials of the second kind and explore some properties. We find generating function of twisted q-Euler polynomials of the second kind. Also, we investigate twisted q-Raabe's multiplication formula and (w, q)-alternating power sums of twisted q-Euler polynomials of the second kind. At the end, we define twisted q-Hurwitz's type Euler zeta function of the second 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 the Korean Mathematical Society
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• v.54 no.4
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• pp.1243-1264
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• 2017

In this paper, we introduce w-Catalan polynomials as a generalization of Catalan polynomials and derive fourteen basic identities of symmetry in three variables related to w-Catalan polynomials and analogues of alternating power sums. In addition, specializations of one of the variables as one give us new and interesting identities of symmetry even for two variables. The derivations of identities are based on the p-adic integral expression for the generating function of the w-Catalan polynomials and the quotient of p-adic integrals for that of the analogues of the alternating power sums.

• Journal of applied mathematics & informatics
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• v.37 no.3_4
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• pp.317-328
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• 2019

This paper defines Carlitz's type (p, q)-Genocchi polynomials and Carlitz's type (h, p, q)-Genocchi polynomials, and explains fourteen properties which can be complemented by Carlitz's type (p, q)-Genocchi polynomials and Carlitz's type (h, p, q)-Genocchi polynomials, including distribution relation, symmetric property, and property of complement. Also, it explores alternating powers sums by proving symmetric property related to Carlitz's type (p, q)-Genocchi polynomials.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1303-1308
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• 2018

In this paper, we consider both maximum modulus and minimum modulus on a circle of some polynomials. These give rise to interesting examples that are about moduli of Chebyshev polynomials and certain sums of polynomials on a circle. Moreover, we obtain some root locations of difference quotients of Chebyshev polynomials.

• Journal of applied mathematics & informatics
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• v.39 no.3_4
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• pp.405-417
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• 2021

The aim of this paper is to provide the residues of Euler polynomials modulo p² in terms of alternating sums of like powers of numbers in arithmetical progression. Also, we establish the analogue of a classical congruence of Lehmer.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.603-618
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• 2015

We derive three identities of symmetry in two variables and eight in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by unramified roots of unity. The case of ramified roots of unity was treated previously. The derivations of identities are based on the p-adic integral expression, with respect to a measure introduced by Koblitz, of the generating function for the generalized twisted Bernoulli polynomials and the quotient of p-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums.

• Kyungpook Mathematical Journal
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• v.63 no.4
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• pp.539-550
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• 2023

In this paper, we define the generalized k-balancing numbers {B^(k)_n} and k-Lucas balancing numbers {C^(k)_n} and associated polynomials, where n is of the form sk+r, 0 ≤ r < k. We give several formulas for these new sequences in terms of classic balancing and Lucas balancing numbers and study their properties. Moreover, we give a Binet style formula, Cassini's identity, and binomial sums of these sequences.

• Journal of the Korean Mathematical Society
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• v.49 no.6
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• pp.1139-1161
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• 2012

Let $p$ be an odd prime number and let F be a finite field with $p^m$ elements. We study representations and strict representations of polynomials $M{\in}F$[T] by sums of ($p^r$ + 1)-th powers. A representation $$M=M_1^k+{\cdots}+M_s^k$$ of $M{\in}F$[T] as a sum of $k$-th powers of polynomials is strict if $k$ deg $M_i<k$ + degM.