• Journal of applied mathematics & informatics
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• v.35 no.5_6
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• pp.587-597
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• 2017

In this paper we define the (p, q)-analogue of Bernoulli numbers and polynomials by generalizing the Bernoulli numbers and polynomials, Carlitz's type q-Bernoulli numbers and polynomials. We also give some interesting properties, explicit formulas, a connection with (p, q)-analogue of Bernoulli numbers and polynomials. Finally, we investigate the zeros of the (p, q)-analogue of Bernoulli polynomials by using computer.

• 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 Korean Mathematical Society
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• v.37 no.3
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• pp.391-410
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• 2000

In the complex case, we construct a q-analogue of the Riemann zeta function q(s) and a q-analogue of the Dirichlet L-function L(s,X), which interpolate the 1-analogue Bernoulli numbers. Using the properties of p-adic integrals and measures, we show that Kummer type congruences for the q-analogue Bernoulli numbers are the generalizations of the usual Kummer congruences for the ordinary Bernoulli numbers. We also construct a q0analogue of the p-adic L-function Lp(s, X;q) which interpolates the q-analogue Bernoulli numbers at non positive integers.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.805-811
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• 2010

It is the aim of this paper to observe an interesting phenomenon of 'scattering' of the zeros of the q-Bernoulli polynomials $B_{n,q}(x)$ for -1 < q < 0 in complex plane. Observe that the structure of the zeros of the Genocchi polynomials $G_n(x)$ resembles the structure of the zeros of the q-Bernoulli polynomials $B_{n,q}(x)$ as q $\rightarrow$ -1.

• Journal of applied mathematics & informatics
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• v.37 no.1_2
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• pp.97-103
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• 2019

In this paper, we give some interesting symmetric identities of the (p, q)-Hurwitz zeta function. We also give some new interesting properties, explicit formulas, a connection with (p, q)-Bernoulli numbers and polynomials.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.399-412
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• 2001

In this paper, we consider that the q-analogue of w$\omega-Bernoulli numbers\; B_i(\omega, q)$. And we calculate the sums of products of two q-analogue of $\omega-Bernoulli numbers B_i(\omega, q)$ in complex cases. From this result, we obtain the Euler type formulas of the Carlitz´s q-Bernoulli numbers $\beta_i(q)$ and q-Bernoulli numbers $B_i(q)$. And we also calculate the p-adic Stirling type series by the definition of $B_i(\omega, q)$ in p-adic cases.

• Journal of applied mathematics & informatics
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• v.35 no.5_6
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• pp.611-621
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• 2017

In this paper, we define a q-analogue of the poly-Bernoulli numbers and polynomials which is generalization of the poly Bernoulli numbers and polynomials including q-polylogarithm function. We also give the relations between generalized poly-Bernoulli polynomials. We derive some relations that are connected with the Stirling numbers of second kind. By using special functions, we investigate some symmetric identities involving q-poly-Bernoulli polynomials.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.205-214
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• 2007

Over the years, there has been increasing interest in solving mathematical problems with the aid of computers. The main purpose of this paper is to investigate the roots of the q-Bernoulli polynomials $B_{n,q}{^r}(x)$ for values of the index n by using computer. Finally, we consider the reflection symmetries of the q-Bernoulli polynomials $B_{n,q}{^r}(x)$.

• Honam Mathematical Journal
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• v.33 no.4
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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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• v.36 no.1_2
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• pp.29-38
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• 2018

In this paper, we introduce a degenerate q-poly-Bernoulli numbers and polynomials include q-logarithm function. We derive some relations with this polynomials and the Stirling numbers of second kind and investigate some symmetric identities using special functions that are involving this polynomials.