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Keywords = q-cosine Bernoulli polynomials

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18 pages, 1242 KiB  
Article
Novel Properties of q-Sine-Based and q-Cosine-Based q-Fubini Polynomials
by Waseem Ahmad Khan, Maryam Salem Alatawi, Cheon Seoung Ryoo and Ugur Duran
Symmetry 2023, 15(2), 356; https://doi.org/10.3390/sym15020356 - 28 Jan 2023
Cited by 6 | Viewed by 1084
Abstract
The main purpose of this paper is to consider q-sine-based and q-cosine-Based q-Fubini polynomials and is to investigate diverse properties of these polynomials. Furthermore, multifarious correlations including q-analogues of the Genocchi, Euler and Bernoulli polynomials, and the q-Stirling [...] Read more.
The main purpose of this paper is to consider q-sine-based and q-cosine-Based q-Fubini polynomials and is to investigate diverse properties of these polynomials. Furthermore, multifarious correlations including q-analogues of the Genocchi, Euler and Bernoulli polynomials, and the q-Stirling numbers of the second kind are derived. Moreover, some approximate zeros of the q-sinebased and q-cosine-Based q-Fubini polynomials in a complex plane are examined, and lastly, these zeros are shown using figures. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)
►▼ Show Figures

Figure 1

Figure 1
<p><math display="inline"><semantics> <mrow> <mrow> <mi>Zeros</mi> <mi>of</mi> </mrow> <msubsup> <mi mathvariant="double-struck">F</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>;</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p><math display="inline"><semantics> <mrow> <mrow> <mi>Zeros</mi> <mi>of</mi> </mrow> <msubsup> <mi mathvariant="double-struck">F</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mn>2</mn> <mo>;</mo> <mn>3</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p><math display="inline"><semantics> <mrow> <mrow> <mi>Zeros</mi> <mi>of</mi> </mrow> <msubsup> <mi mathvariant="double-struck">F</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>;</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p><math display="inline"><semantics> <mrow> <mrow> <mi>Zeros</mi> <mi>of</mi> </mrow> <msubsup> <mi mathvariant="double-struck">F</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>;</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">
18 pages, 1114 KiB  
Article
Diverse Properties and Approximate Roots for a Novel Kinds of the (p,q)-Cosine and (p,q)-Sine Geometric Polynomials
by Sunil Kumar Sharma, Waseem Ahmad Khan, Cheon-Seoung Ryoo and Ugur Duran
Mathematics 2022, 10(15), 2709; https://doi.org/10.3390/math10152709 - 31 Jul 2022
Cited by 3 | Viewed by 1334
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Q-differential/Difference Equations and Related Applications)
►▼ Show Figures

Figure 1

Figure 1
<p>Stacking structure of approximation roots in <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics></math>-cosine geometric polynomials when <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>15</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>25</mn> <mo>,</mo> <mn>30</mn> <mo>,</mo> <mi>p</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Stacking structure of approximation roots in <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics></math>-cosine geometric polynomials when <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>30</mn> <mo>,</mo> <mi>p</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> in 3D.</p> Full article ">Figure 3
<p>Stacking structure of approximation roots in <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics></math>-cosine geometric polynomials when <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>30</mn> <mo>,</mo> <mi>p</mi> <mo>=</mo> <mfrac> <mn>9</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mfrac> <mn>8</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mfrac> <mn>6</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mfrac> <mn>5</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mfrac> <mn>2</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mfrac> <mn>3</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mfrac> <mn>4</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Stacking structure of approximation roots in <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics></math>-sine geometric polynomials when <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>30</mn> <mo>,</mo> <mi>p</mi> <mo>=</mo> <mfrac> <mn>9</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mfrac> <mn>8</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mfrac> <mn>6</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mfrac> <mn>5</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mfrac> <mn>2</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mfrac> <mn>3</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mfrac> <mn>4</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Stacking structure of approximation roots in <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics></math>-sine geometric polynomials when <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>30</mn> <mo>,</mo> <mi>p</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> in 3D.</p> Full article ">
12 pages, 290 KiB  
Article
On (p, q)-Sine and (p, q)-Cosine Fubini Polynomials
by Waseem Ahmad Khan, Ghulam Muhiuddin, Ugur Duran and Deena Al-Kadi
Symmetry 2022, 14(3), 527; https://doi.org/10.3390/sym14030527 - 4 Mar 2022
Cited by 6 | Viewed by 1937
Abstract
In recent years, (p,q)-special polynomials, such as p,q-Euler, p,q-Genocchi, p,q-Bernoulli, and p,q-Frobenius-Euler, have been studied and investigated by many mathematicians, as well physicists. It is important [...] Read more.
In recent years, (p,q)-special polynomials, such as p,q-Euler, p,q-Genocchi, p,q-Bernoulli, and p,q-Frobenius-Euler, have been studied and investigated by many mathematicians, as well physicists. It is important that any polynomial have explicit formulas, symmetric identities, summation formulas, and relations with other polynomials. In this work, the (p,q)-sine and (p,q)-cosine Fubini polynomials are introduced and multifarious abovementioned properties for these polynomials are derived by utilizing some series manipulation methods. p,q-derivative operator rules and p,q-integral representations for the (p,q)-sine and (p,q)-cosine Fubini polynomials are also given. Moreover, several correlations related to both the (p,q)-Bernoulli, Euler, and Genocchi polynomials and the (p,q)-Stirling numbers of the second kind are developed. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)
21 pages, 939 KiB  
Article
Structure of Approximate Roots Based on Symmetric Properties of (p, q)-Cosine and (p, q)-Sine Bernoulli Polynomials
by Cheon Seoung Ryoo and Jung Yoog Kang
Symmetry 2020, 12(6), 885; https://doi.org/10.3390/sym12060885 - 30 May 2020
Cited by 3 | Viewed by 1951
Abstract
This paper constructs and introduces ( p , q ) -cosine and ( p , q ) -sine Bernoulli polynomials using ( p , q ) -analogues of ( x + a ) n . Based on these polynomials, we discover basic properties [...] Read more.
This paper constructs and introduces ( p , q ) -cosine and ( p , q ) -sine Bernoulli polynomials using ( p , q ) -analogues of ( x + a ) n . Based on these polynomials, we discover basic properties and identities. Moreover, we determine special properties using ( p , q ) -trigonometric functions and verify various symmetric properties. Finally, we check the symmetric structure of the approximate roots based on symmetric polynomials. Full article
(This article belongs to the Special Issue Polynomials: Special Polynomials and Number-Theoretical Applications)
►▼ Show Figures

Figure 1

Figure 1
<p>Stacking structure of approximate roots in the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics></math>-cosine Bernoulli polynomials when <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Stacking structure of approximate roots in the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics></math>-cosine Bernoulli polynomials when <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Stacking structure of approximate roots in the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics></math>-sine Bernoulli polynomials when <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> in 3D.</p> Full article ">Figure 4
<p>Stacking structure of approximate roots in the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics></math>-sine Bernoulli polynomials when <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> in 3D.</p> Full article ">
18 pages, 14803 KiB  
Article
Various Structures of the Roots and Explicit Properties of q-cosine Bernoulli Polynomials and q-sine Bernoulli Polynomials
by Jung Yoog Kang and Chen Seoung Ryoo
Mathematics 2020, 8(4), 463; https://doi.org/10.3390/math8040463 - 25 Mar 2020
Cited by 12 | Viewed by 2455
Abstract
In this paper, we define cosine Bernoulli polynomials and sine Bernoulli polynomials related to the q-number. Furthermore, we intend to find the properties of these polynomials and check the structure of the roots. Through numerical experimentation, we look for various assumptions about [...] Read more.
In this paper, we define cosine Bernoulli polynomials and sine Bernoulli polynomials related to the q-number. Furthermore, we intend to find the properties of these polynomials and check the structure of the roots. Through numerical experimentation, we look for various assumptions about the polynomials above. Full article
(This article belongs to the Special Issue Special Functions and Applications)
►▼ Show Figures

Figure 1

Figure 1
<p>Approximate root of <math display="inline"><semantics> <mrow> <msub> <mrow/> <mi>C</mi> </msub> <msub> <mi>B</mi> <mrow> <mi>n</mi> <mo>,</mo> <mn>0.9</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>10</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>30</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Approximate roots for <math display="inline"><semantics> <mrow> <msub> <mrow/> <mi>C</mi> </msub> <msub> <mi>B</mi> <mrow> <mn>50</mn> <mo>,</mo> <mn>0.9</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Approximate roots of <math display="inline"><semantics> <mrow> <msub> <mrow/> <mi>C</mi> </msub> <msub> <mi>B</mi> <mrow> <mi>n</mi> <mo>,</mo> <mn>0.5</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>10</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>30</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Stacking structure in 3D of <math display="inline"><semantics> <mrow> <msub> <mrow/> <mi>C</mi> </msub> <msub> <mi>B</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>30</mn> <mo>,</mo> <mn>0.5</mn> <mo>≤</mo> <mi>q</mi> <mo>≤</mo> <mn>0.9</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Approximate roots of <math display="inline"><semantics> <mrow> <msub> <mrow/> <mi>C</mi> </msub> <msub> <mi>B</mi> <mrow> <mi>n</mi> <mo>,</mo> <mn>0.1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>10</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>30</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Stacking structure of <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>30</mn> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mn>0.9</mn> <mo>,</mo> <mn>0.5</mn> <mo>,</mo> <mn>0.1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Stacking structure of <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>30</mn> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mn>0.9</mn> <mo>,</mo> <mn>0.5</mn> <mo>,</mo> <mn>0.1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Stacking structure of <math display="inline"><semantics> <mrow> <msub> <mrow/> <mi>S</mi> </msub> <msub> <mi>B</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>30</mn> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mn>0.9</mn> <mo>,</mo> <mn>0.5</mn> <mo>,</mo> <mn>0.1</mn> </mrow> </semantics></math>.</p> Full article ">
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