• 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.51 no.5
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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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• v.41 no.2
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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.

• 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.

• 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.38 no.1_2
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• pp.145-157
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• 2020

In this article, we define a generating function of the generalized (p, q)-poly-Bernoulli polynomials with variable a by using the polylogarithm function. From the definition, we derive some properties that is concerned with other numbers and polynomials. Furthermore, we construct a special functions and give some symmetric identities involving the generalized (p, q)-poly-Bernoulli polynomials and power sums of the first integers.

• Journal of applied mathematics & informatics
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• v.38 no.5_6
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• pp.601-609
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• 2020

In this paper, we introduce degenerate generalized poly-Bernoulli numbers and polynomials with (p, q)-logarithm function. We find some identities that are concerned with the Stirling numbers of second kind and derive symmetric identities by using generalized falling factorial sum.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1149-1156
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• 2016

In this paper, we introduce the degenerate Carlitz q-Bernoulli numbers and polynomials and give some interesting identities and properties of these numbers and polynomials which are derived from the generating functions and p-adic integral equations.

• East Asian mathematical journal
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• v.36 no.1
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• pp.13-24
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• 2020

The main purpose of this paper is to introduce Apostol type (p, q)-Frobenius-Genocchi numbers and polynomials of order α and investigate some basic identities and properties for these polynomials and numbers including addition theorems, difference equations, derivative properties, recurrence relations. We also obtain integral representations, implicit and explicit formulas and relations for these polynomials and numbers. Furthermore, we consider some relationships for Apostol type (p, q)-Frobenius-Genocchi polynomials of order α associated with (p, q)-Apostol Bernoulli polynomials, (p, q)-Apostol Euler polynomials and (p, q)-Apostol Genocchi polynomials.

• Journal of applied mathematics & informatics
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• v.37 no.3_4
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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.