Bivariate Generalized Shifted Gegenbauer Orthogonal System
For K0,K1≥0, λ>−1/2, we examine Cr∗λ,K0,K1x, generalized shifted Gegenbauer orthogonal polynomials, with reference to the weight Wλ,K0,K1x=2λΓ2λ/Γλ+1/22x−x2λ−1/2Ix∈0,1dx+K0δ0+K1δ1, where the indicator function is denoted by Ix∈0,1 and δx indicates the Dirac δ−measure. Then, we construct a bivaria...
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2021-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/5563032 |
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| author | Mohammad A. Alqudah Maalee N. Almheidat Tareq Hamadneh |
| author_facet | Mohammad A. Alqudah Maalee N. Almheidat Tareq Hamadneh |
| author_sort | Mohammad A. Alqudah |
| collection | DOAJ |
| description | For K0,K1≥0, λ>−1/2, we examine Cr∗λ,K0,K1x, generalized shifted Gegenbauer orthogonal polynomials, with reference to the weight Wλ,K0,K1x=2λΓ2λ/Γλ+1/22x−x2λ−1/2Ix∈0,1dx+K0δ0+K1δ1, where the indicator function is denoted by Ix∈0,1 and δx indicates the Dirac δ−measure. Then, we construct a bivariate generalized shifted Gegenbauer orthogonal system ℭn,r,d∗λ,K0,K1u,v,w over a triangular domain T, with reference to a bivariate measure Wλ,γ,K0,K1u,v,w=Γ2λ+1/Γλ+1/22uλ−1/21−vλ−1/21−wγ−1Iu∈0,1−wIw∈0,1dudw+K0δ0u+K1δw−1u, which is given explicitly in the Bézier form as ℭn,r,d∗λ,K0,K1u,v,w=∑i+j+k=nai,j,kn,r,dBi,j,knu,v,w. In addition, for d=0,…,k, r=0,1,…,n, and n∈0∪ℕ, we write the coefficients ai,j,kn,r,d in closed form and produce an equation that generates the coefficients recursively. |
| format | Article |
| id | doaj-art-b45181c6b94f45718db40da631812204 |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-b45181c6b94f45718db40da6318122042025-02-03T01:28:26ZengWileyJournal of Mathematics2314-46292314-47852021-01-01202110.1155/2021/55630325563032Bivariate Generalized Shifted Gegenbauer Orthogonal SystemMohammad A. Alqudah0Maalee N. Almheidat1Tareq Hamadneh2German Jordanian University, Amman 11180, JordanDepartment of Mathematics, University of Petra, Amman 11196, JordanDepartment of Mathematics, Al-Zaytoonah University of Jordan, P. O. Box 130, Amman, JordanFor K0,K1≥0, λ>−1/2, we examine Cr∗λ,K0,K1x, generalized shifted Gegenbauer orthogonal polynomials, with reference to the weight Wλ,K0,K1x=2λΓ2λ/Γλ+1/22x−x2λ−1/2Ix∈0,1dx+K0δ0+K1δ1, where the indicator function is denoted by Ix∈0,1 and δx indicates the Dirac δ−measure. Then, we construct a bivariate generalized shifted Gegenbauer orthogonal system ℭn,r,d∗λ,K0,K1u,v,w over a triangular domain T, with reference to a bivariate measure Wλ,γ,K0,K1u,v,w=Γ2λ+1/Γλ+1/22uλ−1/21−vλ−1/21−wγ−1Iu∈0,1−wIw∈0,1dudw+K0δ0u+K1δw−1u, which is given explicitly in the Bézier form as ℭn,r,d∗λ,K0,K1u,v,w=∑i+j+k=nai,j,kn,r,dBi,j,knu,v,w. In addition, for d=0,…,k, r=0,1,…,n, and n∈0∪ℕ, we write the coefficients ai,j,kn,r,d in closed form and produce an equation that generates the coefficients recursively.http://dx.doi.org/10.1155/2021/5563032 |
| spellingShingle | Mohammad A. Alqudah Maalee N. Almheidat Tareq Hamadneh Bivariate Generalized Shifted Gegenbauer Orthogonal System Journal of Mathematics |
| title | Bivariate Generalized Shifted Gegenbauer Orthogonal System |
| title_full | Bivariate Generalized Shifted Gegenbauer Orthogonal System |
| title_fullStr | Bivariate Generalized Shifted Gegenbauer Orthogonal System |
| title_full_unstemmed | Bivariate Generalized Shifted Gegenbauer Orthogonal System |
| title_short | Bivariate Generalized Shifted Gegenbauer Orthogonal System |
| title_sort | bivariate generalized shifted gegenbauer orthogonal system |
| url | http://dx.doi.org/10.1155/2021/5563032 |
| work_keys_str_mv | AT mohammadaalqudah bivariategeneralizedshiftedgegenbauerorthogonalsystem AT maaleenalmheidat bivariategeneralizedshiftedgegenbauerorthogonalsystem AT tareqhamadneh bivariategeneralizedshiftedgegenbauerorthogonalsystem |