• 대한수학회지
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• 제51권5호
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• pp.1045-1073
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• 2014

The study of the identities of symmetry for the Bernoulli polynomials arises from the study of Gauss's multiplication formula for the gamma function. There are many works in this direction. In the sense of p-adic analysis, the q-Bernoulli polynomials are natural extensions of the Bernoulli and Apostol-Bernoulli polynomials (see the introduction of this paper). By using the N-fold iterated Volkenborn integral, we derive serval identities of symmetry related to the q-extension power sums and the higher order q-Bernoulli polynomials. Many previous results are special cases of the results presented in this paper, including Tuenter's classical results on the symmetry relation between the power sum polynomials and the Bernoulli numbers in [A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001), no. 3, 258-261] and D. S. Kim's eight basic identities of symmetry in three variables related to the q-analogue power sums and the q-Bernoulli polynomials in [Identities of symmetry for q-Bernoulli polynomials, Comput. Math. Appl. 60 (2010), no. 8, 2350-2359].

• Journal of applied mathematics & informatics
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• 제41권2호
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• pp.439-448
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• 2023

We introduce several higher-order (p, q)-differential equation of which are related to (p, q)-Bernoulli polynomials. We also find some relations between (p, q)-Bernoulli, (p, q)-Euler, and (p, q)-Genocchi polynomials.

• 호남수학학술지
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• 제33권4호
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• pp.519-527
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• 2011

Recently, the modified q-Bernoulli numbers and polynomials with weight ${\alpha}$ are introduced in [3]: In this paper we give some interesting p-adic integral representation on $\mathbb{Z}_p$ of the weighted q-Bernstein polynomials related to the modified q-Bernoulli numbers and polynomials with weight ${\alpha}$. From those integral representation on $\mathbb{Z}_p$ of the weighted q-Bernstein polynomials, we can derive some identities on the modified q-Bernoulli numbers and polynomials with weight ${\alpha}$.

• Journal of applied mathematics & informatics
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• 제37권3_4호
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• pp.295-305
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• 2019

In the present paper, we introduce (p, q)-Frobenius-Genocchi numbers and polynomials and investigate some basic identities and properties for these polynomials and numbers including addition theorems, difference equations, derivative properties, recurrence relations and so on. Then, we provide integral representations, implicit and explicit formulas and relations for these polynomials and numbers. We consider some relationships for (p, q)-Frobenius-Genocchi polynomials of order ${\alpha}$ associated with (p, q)-Bernoulli polynomials, (p, q)-Euler polynomials and (p, q)-Genocchi polynomials.

• 호남수학학술지
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• 제34권1호
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• pp.1-9
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• 2012

The purpose of this paper is to give a new construction of the extended q-Euler numbers and polynomials of higher-order with weight by using p-adic q-integral on $\mathbb{Z}_p$.

• 대한수학회지
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• 제56권1호
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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.