A Numerical Computation of Zeros of q-Generalized Tangent-Appell Polynomials
Abstract
:1. Introduction and Preliminaries
2. q-Generalized Tangent-Appell Polynomials
3. Identities Involving q-Generalized Tangent-Appell Polynomials
4. Graphical Representation and Computation of Zeros
Author Contributions
Funding
Conflicts of Interest
References
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Name of q-Special | ||||
---|---|---|---|---|
S. No. | Polynomials and Its | Generating Function | Series Definition | |
Associated Numbers | ||||
I | q-Bernoulli polynomials; | |||
q-Bernoulli numbers | ||||
[11,12] | ||||
II | q-Euler polynomials; | |||
q-Euler numbers | ||||
[6,12] |
S. No. | Name of the | Generating Function | Generating Function | |
---|---|---|---|---|
Resultant Member | of Resultant Polynomial | of Resultant Number | ||
I | q-generalized tangent | |||
-Bernoulli polynomials (qGTBP) | ||||
II | q-generalized tangent | |||
-Euler polynomials (qGTEP) |
S. No. | Results | qGTBP | qGTEP |
---|---|---|---|
I | Series | ||
Expansions | |||
II | Summation | ||
Formulae | |||
III | Differential | ||
Recurrence | |||
Relations | |||
S. No. | Identities Involving q-Generalized Tangent | Identities Involving q-Generalized Tangent |
---|---|---|
-Bernoulli Polynomials qGTBP | -Euler Polynomials qGTEP | |
I | ||
II | ||
III | ||
IV | ||
V | ||
VI | ||
VII | ||
Degree n | Number of Real Zeros | Number of Complex Zeros |
---|---|---|
1 | 1 | 0 |
2 | 2 | 0 |
3 | 3 | 0 |
4 | 2 | 2 |
5 | 3 | 2 |
6 | 2 | 4 |
7 | 3 | 4 |
8 | 2 | 6 |
9 | 3 | 6 |
Degree n | u |
---|---|
1 | 1.0000 |
2 | −2.2609, 2.9276 |
3 | −1.8173, −0.15699, 2.7521 |
4 | −2.1632, 2.9221 |
5 | −1.7892, −1.2772, 2.9430 |
6 | −2.1077, 2.9725 |
7 | 1.8001, −1.5784, 2.9835 |
8 | −2.0768, 2.9910 |
9 | −1.8504, −1.6667, 2.9948 |
Degree n | Number of Real Zeros | Number of Complex Zeros |
---|---|---|
1 | 1 | 0 |
2 | 2 | 0 |
3 | 3 | 0 |
4 | 2 | 2 |
5 | 3 | 2 |
6 | 4 | 2 |
7 | 5 | 2 |
8 | 4 | 4 |
9 | 5 | 4 |
10 | 2 | 8 |
Degree n | u |
---|---|
1 | 0 |
2 | −2.1602, 2.1602 |
3 | −2.1498, −0.73464, 2.8845 |
4 | 0.30406, 3.0659 |
5 | −2.0953, 1.0226, 3.0658 |
6 | −1.9820, 1.4174, 3.0287 |
7 | −1.9583, −1.2749, −0.42473, 1.5804, 3.0043 |
8 | −2.0338, 0.48493, 1.5889, 2.9970 |
9 | −1.9779, −1.4188, 1.0634, 1.4592, 2.9975 |
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Yasmin, G.; Ryoo, C.S.; Islahi, H. A Numerical Computation of Zeros of q-Generalized Tangent-Appell Polynomials. Mathematics 2020, 8, 383. https://doi.org/10.3390/math8030383
Yasmin G, Ryoo CS, Islahi H. A Numerical Computation of Zeros of q-Generalized Tangent-Appell Polynomials. Mathematics. 2020; 8(3):383. https://doi.org/10.3390/math8030383
Chicago/Turabian StyleYasmin, Ghazala, Cheon Seoung Ryoo, and Hibah Islahi. 2020. "A Numerical Computation of Zeros of q-Generalized Tangent-Appell Polynomials" Mathematics 8, no. 3: 383. https://doi.org/10.3390/math8030383
APA StyleYasmin, G., Ryoo, C. S., & Islahi, H. (2020). A Numerical Computation of Zeros of q-Generalized Tangent-Appell Polynomials. Mathematics, 8(3), 383. https://doi.org/10.3390/math8030383