New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas
Abstract
:1. Introduction
- An approach based on deriving a new formula of the moments of the shifted normalized Jacobi polynomials in terms of their original shifted Jacobi polynomials but with different parameters;
- An approach based on making use of the connection formulas between two different normalized Jacobi polynomials.
- In the articles [28,31,34], the linearization formulas were established by reducing some exiting ones in the literature with the aid of some celebrated reduction formulas or via some symbolic algorithms; however, in the current article, we establish two new approaches for deriving some linearization formulas, and after that reduce these linearization formulas by symbolic computation.
- The articles [9,29,30] deal with some special linearization formulas. In fact, the approaches followed were based on expressing products of hypergeometric functions in terms of a single generalized hypergeometric function using some suitable transformation formulas; however, the current article deals with some general linearization formulas.
- We do believe that the approach based on the moments formulas can be followed to establish linearization formulas of different orthogonal polynomials and not restricted to Jacobi polynomials.
2. Some Elementary Properties of the Classical Jacobi Polynomials and Their Shifted Ones
3. New Moments Formulas of the Shifted Normalized Jacobi Polynomials
4. A New Approach for Solving Jacobi Linearization Problem via Moments Formulas
5. Some New Linearization Formulas of Chebyshev Polynomials Using the Connection Coefficients Approach
6. Numerical Application on the Non-Linear Riccati Equation
6.1. Tau Algorithm for the Non-Linear Riccati Differential Equation
6.2. Numerical Tests
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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N | 6 | 8 | 10 | 12 | 14 | 16 |
---|---|---|---|---|---|---|
E | 2.358 × | 3.264 × | 6.382 × | 5.943 × | 2.975 × | 6.241 × |
x | OHAM [48] | MHPM [49] | VIM [50] | IRKHSM [51] | The Method in [9] | |
---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 |
0.1 | 3.20 | 1.00 | 1.98 | 3.58 | 1.52 | 1.27 |
0.2 | 2.90 | 1.20 | 1.03 | 7.58 | 1.27 | 2.23 |
0.3 | 1.10 | 1.00 | 8.85 | 1.20 | 2.57 | 1.34 |
0.4 | 2.50 | 3.03 | 3.33 | 1.66 | 3.27 | 2.31 |
0.5 | 4.40 | 1.55 | 7.26 | 2.12 | 3.57 | 3.68 |
0.6 | 5.50 | 4.69 | 9.98 | 2.52 | 4.15 | 4.32 |
0.7 | 5.50 | 1.05 | 8.84 | 2.87 | 4.21 | 4.95 |
0.8 | 3.80 | 1.88 | 1.54 | 3.40 | 4.31 | 5.62 |
0.9 | 3.20 | 2.80 | 4.99 | 4.90 | 4.35 | 5.94 |
1.0 | 3.40 | 3.43 | 3.47 | 9.22 | 4.42 | 6.24 |
N | 8 | 10 | 12 | 14 | 16 |
---|---|---|---|---|---|
3.51 | 2.45 | 2.39 | 5.94 | 5.37 | |
1.38 | 3.27 | 8.36 | 4.62 | 3.74 | |
4.58 | 3.19 | 4.94 | 6.84 | 2.22 |
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Abd-Elhameed, W.M.; Badah, B.M. New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas. Mathematics 2021, 9, 1573. https://doi.org/10.3390/math9131573
Abd-Elhameed WM, Badah BM. New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas. Mathematics. 2021; 9(13):1573. https://doi.org/10.3390/math9131573
Chicago/Turabian StyleAbd-Elhameed, Waleed Mohamed, and Badah Mohamed Badah. 2021. "New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas" Mathematics 9, no. 13: 1573. https://doi.org/10.3390/math9131573
APA StyleAbd-Elhameed, W. M., & Badah, B. M. (2021). New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas. Mathematics, 9(13), 1573. https://doi.org/10.3390/math9131573