Explicit Properties of q-Cosine and q-Sine Euler Polynomials Containing Symmetric Structures

Ryoo, C. S.; Kang, Ji-Young

doi:10.3390/sym12081247

Cited by 7 publications

(3 citation statements)

References 18 publications

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“…where the higher-order hypergeometric q-Bernoulli polynomials are provided by (see Ryoo et al, 2020) e q (ψz)…”

Section: Discussionmentioning

confidence: 99%

CONSTRUCTION FOR q-HYPERGEOMETRIC BERNOULLI POLYNOMIALS OF A COMPLEX VARIABLE WITH APPLICATIONS COMPUTER MODELING

Al edamat

2024

AMMA

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Two new extensions of the familiar Bernoulli polynomials are considered by using q-sine, q-cosine, q-hypergeometric and q-exponential functions. We call q-sine and q-cosine hypergeometric Bernoulli polynomials. Then, diverse formulas and properties for these polynomials, such as summation formulas, addition formulas, q-derivative properties, q-integral representations and some correlations are derived. Also, q-sine and q-cosine hypergeometric Bernoulli polynomials with two parameters are introduced and some relations and identities are investigated. Furthermore, some computational values are given by tables, and the beautiful zeros representations of the q-sine hypergeometric Bernoulli polynomials and q-cosine hypergeometric Bernoulli polynomials are showed by the figures

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“…where the higher-order hypergeometric q-Bernoulli polynomials are provided by (see Ryoo et al, 2020) e q (ψz)…”

Section: Discussionmentioning

confidence: 99%

CONSTRUCTION FOR q-HYPERGEOMETRIC BERNOULLI POLYNOMIALS OF A COMPLEX VARIABLE WITH APPLICATIONS COMPUTER MODELING

Al edamat

2024

AMMA

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“…By making use of q-numbers and q-concepts, Jang et al [2,4] defined q-Bernoulli polynomials and numbers, q-Genocchi polynomials and numbers and q-Euler polynomials and numbers and provided some new and interesting identities and formulae. With this viewpoint, several authors have introduced q-analogues of special numbers and polynomials and have investigated their properties.…”

Section: Discussionmentioning

confidence: 99%

“…Recently, many authors have considered and applied the generating functions techniques to new families of special polynomials, including two parametric kinds of polynomials, such as Bernoulli, Euler, Genocchi, etc. (see [1][2][3][4][5][6][7][8][9][10]). They have firstly derived the basic identities of these polynomials.…”

Section: Introductionmentioning

confidence: 99%

Explicit Properties of Apostol-Type Frobenius–Euler Polynomials Involving q-Trigonometric Functions with Applications in Computer Modeling

et al. 2023

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In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. By using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. By making use of a partial derivative operator, we derived some interesting finite combinatorial sums. Finally, we detail some special cases for these results.

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Diverse Properties and Approximate Roots for a Novel Kinds of the (p,q)-Cosine and (p,q)-Sine Geometric Polynomials

et al. 2022

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Utilizing p,q-numbers and p,q-concepts, in 2016, Duran et al. considered p,q-Genocchi numbers and polynomials, p,q-Bernoulli numbers and polynomials and p,q-Euler polynomials and numbers and provided multifarious formulas and properties for these polynomials. Inspired and motivated by this consideration, many authors have introduced (p,q)-special polynomials and numbers and have described some of their properties and applications. In this paper, using the (p,q)-cosine polynomials and (p,q)-sine polynomials, we consider a novel kinds of (p,q)-extensions of geometric polynomials and acquire several properties and identities by making use of some series manipulation methods. Furthermore, we compute the p,q-integral representations and p,q-derivative operator rules for the new polynomials. Additionally, we determine the movements of the approximate zerosof the two mentioned polynomials in a complex plane, utilizing the Newton method, and we illustrate them using figures.

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Explicit Properties of q-Cosine and q-Sine Euler Polynomials Containing Symmetric Structures

Cited by 7 publications

References 18 publications

CONSTRUCTION FOR q-HYPERGEOMETRIC BERNOULLI POLYNOMIALS OF A COMPLEX VARIABLE WITH APPLICATIONS COMPUTER MODELING

CONSTRUCTION FOR q-HYPERGEOMETRIC BERNOULLI POLYNOMIALS OF A COMPLEX VARIABLE WITH APPLICATIONS COMPUTER MODELING

Explicit Properties of Apostol-Type Frobenius–Euler Polynomials Involving q-Trigonometric Functions with Applications in Computer Modeling

Diverse Properties and Approximate Roots for a Novel Kinds of the (p,q)-Cosine and (p,q)-Sine Geometric Polynomials

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