• 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.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.29-39
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• 2015

In this paper, we investigate the functional equations of the multiple Dirichlet and Hurwitz L-functions associated with Bernoulli numbers and polynomials attached to Dirichlet character.

• Kyungpook Mathematical Journal
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• v.50 no.2
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• pp.229-235
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• 2010

In the present paper, we obtain a new formula for the generalized Weyl differintegral operator in a compact form avoiding the occurrence of infinite series and thus making it useful in applications. Our findings provide interesting generalizations and unifications of the results given by several authors and lying scattered in the literature.

• Kyungpook Mathematical Journal
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• v.47 no.4
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• pp.495-500
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• 2007

The aim of this paper is to derive a fractional derivative of the multivariable H-function of Srivastava and Panda [7], associated with a general class of multivariable polynomials of Srivastava [4] and the generalized Lauricella functions of Srivastava and Daoust [9]. Certain special cases have also been discussed. The results derived here are of a very general nature and hence encompass several cases of interest hitherto scattered in the literature.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.651-669
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• 2018

The article is themed to classify new (fully) degenerate Hermite-Bernoulli polynomials with formulation in terms of p-adic fermionic integrals on $\mathbb{Z}_p$. The entire paper is designed to illustrate new properties in association with Daehee polynomials in a consolidated and generalized form.

• Kyungpook Mathematical Journal
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• v.64 no.1
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• pp.133-159
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• 2024

In a previous work we studied generalized Stirling numbers of the second kind S^{(a₂,b₂,p₂)}_a1,b1 (p₁, k), where a₁, a₂, b₁, b₂ are given complex numbers, a₁, a₂ ≠ 0, and p₁, p₂ are non-negative integers given. In this work we use these generalized Stirling numbers to define Bernoulli polynomials in two variables B_p1,p2 (x₁, x₂), and Euler polynomials in two variables E_p1p2 (x₁, x₂). By using results for S^{(1,x₂,p₂)}_1,x1 (p₁, k), we obtain generalizations, to the bivariate case, of some well-known properties from the standard case, as addition formulas, difference equations and sums of powers. We obtain some identities for bivariate Bernoulli and Euler polynomials, and some generalizations, to the bivariate case, of several known identities for Bernoulli and Euler numbers and polynomials of the standard case.

• 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.

• Nonlinear Functional Analysis and Applications
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• v.27 no.1
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• pp.169-187
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• 2022

There have been provided a surprisingly large number of summation formulae for generalized hypergeometric functions and series incorporating a variety of elementary and special functions in their various combinations. In this paper, we aim to consider certain generalized hypergeometric function ₃F₂ with particular arguments, through which a number of summation formulas for _p+1F_p(1) are provided. We then establish a power series whose coefficients are involved in generalized hypergeometric functions with unit argument. Also, we demonstrate that the generalized hypergeometric functions with unit argument mentioned before may be expressed in terms of Bell polynomials. Further, we explore several special instances of our primary identities, among numerous others, and raise a problem that naturally emerges throughout the course of this investigation.

• Journal of the Korean Mathematical Society
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• v.48 no.5
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• pp.913-926
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• 2011

The family of q-Laguerre polynomials $\{L_n^{(\alpha)}({\cdot};q)\}_{n=0}^{\infty}$ is usually defined for 0 < q < 1 and ${\alpha}$ > -1. We extend this family to a new one in which arbitrary complex values of the parameter ${\alpha}$ are allowed. These so-called generalized q-Laguerre polynomials fulfil the same three term recurrence relation as the original ones, but when the parameter ${\alpha}$ is a negative integer, no orthogonality property can be deduced from Favard's theorem. In this work we introduce non-standard inner products involving q-derivatives with respect to which the generalized q-Laguerre polynomials $\{L_n^{(-N)}({\cdot};q)\}_{n=0}^{\infty}$, for positive integers N, become orthogonal.

• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.695-708
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• 2022

In this paper, we present a new definition and generalization of the first and second kinds of Daehee numbers and polynomials with the higher order. Some new results for these polynomials and numbers are derived. Furthermore, some interesting special cases of the new generalized Daehee polynomials and numbers of higher order are deduced.