• Honam Mathematical Journal
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• v.33 no.2
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• pp.173-179
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• 2011

The main aim of this paper is to give some relationships between q-Bernstein, higher order genocchi and Euler polynomials.

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.491-499
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• 2005

Recently, Kim[2,6] has introduced an interesting Euler­Barnes' numbers and polynomials. In this paper, we construct the q-analogue of Euler-Barnes' numbers and polynomials, and investigate their properties.

• 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.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.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.435-453
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• 2008

In this paper, by using q-deformed bosonic p-adic integral, we give $\lambda$-Bernoulli numbers and polynomials, we prove Witt's type formula of $\lambda$-Bernoulli polynomials and Gauss multiplicative formula for $\lambda$-Bernoulli polynomials. By using derivative operator to the generating functions of $\lambda$-Bernoulli polynomials and generalized $\lambda$-Bernoulli numbers, we give Hurwitz type $\lambda$-zeta functions and Dirichlet's type $\lambda$-L-functions; which are interpolated $\lambda$-Bernoulli polynomials and generalized $\lambda$-Bernoulli numbers, respectively. We give generating function of $\lambda$-Bernoulli numbers with order r. By using Mellin transforms to their function, we prove relations between multiply zeta function and $\lambda$-Bernoulli polynomials and ordinary Bernoulli numbers of order r and $\lambda$-Bernoulli numbers, respectively. We also study on $\lambda$-Bernoulli numbers and polynomials in the space of locally constant. Moreover, we define $\lambda$-partial zeta function and interpolation function.