New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas
This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi...
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MDPI AG
2021-07-01
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| Online Access: | https://www.mdpi.com/2227-7390/9/13/1573 |
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| author | Waleed Mohamed Abd-Elhameed Badah Mohamed Badah |
| author_facet | Waleed Mohamed Abd-Elhameed Badah Mohamed Badah |
| author_sort | Waleed Mohamed Abd-Elhameed |
| collection | DOAJ |
| description | This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mspace width="4pt"></mspace><mn>4</mn><msub><mi>F</mi><mn>3</mn></msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method. |
| first_indexed | 2024-03-10T09:50:44Z |
| format | Article |
| id | doaj.art-590adf2872c54db28344bbbb932b9580 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T09:50:44Z |
| publishDate | 2021-07-01 |
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| spelling | doaj.art-590adf2872c54db28344bbbb932b95802023-11-22T02:46:27ZengMDPI AGMathematics2227-73902021-07-01913157310.3390/math9131573New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection FormulasWaleed Mohamed Abd-Elhameed0Badah Mohamed Badah1Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, EgyptDepartment of Mathematics, College of Science, University of Jeddah, Jeddah 23218, Saudi ArabiaThis article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mspace width="4pt"></mspace><mn>4</mn><msub><mi>F</mi><mn>3</mn></msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.https://www.mdpi.com/2227-7390/9/13/1573Jacobi polynomialsgeneralized hypergeometric functionsChebyshev polynomialslinearization coefficientsconnection formulasmoments formulas |
| spellingShingle | Waleed Mohamed Abd-Elhameed Badah Mohamed Badah New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas Mathematics Jacobi polynomials generalized hypergeometric functions Chebyshev polynomials linearization coefficients connection formulas moments formulas |
| title | New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas |
| title_full | New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas |
| title_fullStr | New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas |
| title_full_unstemmed | New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas |
| title_short | New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas |
| title_sort | new approaches to the general linearization problem of jacobi polynomials based on moments and connection formulas |
| topic | Jacobi polynomials generalized hypergeometric functions Chebyshev polynomials linearization coefficients connection formulas moments formulas |
| url | https://www.mdpi.com/2227-7390/9/13/1573 |
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