On generalized Heun equation with some mathematical properties
We study the analytic solutions of the generalized Heun equation, (α0 + α1 r + α2 r2 + α3 r3) y′′ + (β0 + β1 r + β2 r2) y′ + (ε0 + ε1 r) y = 0, where |α3| + |β2|≠ 0, and {αi}3i=0, {βi}2i=0, {εi}1i=0 are real parameters. The existence conditions for the polynomial solutions are given. A simple proced...
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| Format: | Article |
| Language: | English |
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CTU Central Library
2022-02-01
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| Series: | Acta Polytechnica |
| Subjects: | |
| Online Access: | https://ojs.cvut.cz/ojs/index.php/ap/article/view/7583 |
| _version_ | 1811272752907157504 |
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| author | Nasser Saad |
| author_facet | Nasser Saad |
| author_sort | Nasser Saad |
| collection | DOAJ |
| description | We study the analytic solutions of the generalized Heun equation, (α0 + α1 r + α2 r2 + α3 r3) y′′ + (β0 + β1 r + β2 r2) y′ + (ε0 + ε1 r) y = 0, where |α3| + |β2|≠ 0, and {αi}3i=0, {βi}2i=0, {εi}1i=0 are real parameters. The existence conditions for the polynomial solutions are given. A simple procedure based on a recurrence relation is introduced to evaluate these polynomial solutions explicitly. For α0 = 0, α1≠ 0, we prove that the polynomial solutions of the corresponding differential equation are sources of finite sequences of orthogonal polynomials. Several mathematical properties, such as the recurrence relation, Christoffel-Darboux formulas and the norms of these polynomials, are discussed. We shall also show that they exhibit a factorization property that permits the construction of other infinite sequences of orthogonal polynomials. |
| first_indexed | 2024-04-12T22:45:56Z |
| format | Article |
| id | doaj.art-5f8cd9031c5347a8b20d80fc6c89f7e5 |
| institution | Directory Open Access Journal |
| issn | 1805-2363 |
| language | English |
| last_indexed | 2024-04-12T22:45:56Z |
| publishDate | 2022-02-01 |
| publisher | CTU Central Library |
| record_format | Article |
| series | Acta Polytechnica |
| spelling | doaj.art-5f8cd9031c5347a8b20d80fc6c89f7e52022-12-22T03:13:31ZengCTU Central LibraryActa Polytechnica1805-23632022-02-0162116518910.14311/AP.2022.62.01654823On generalized Heun equation with some mathematical propertiesNasser Saad0University of Prince Edward Island, Department of Mathematics and Statistics, 550 University Avenue, Charlottetown, PEI, Canada C1A 4P3.We study the analytic solutions of the generalized Heun equation, (α0 + α1 r + α2 r2 + α3 r3) y′′ + (β0 + β1 r + β2 r2) y′ + (ε0 + ε1 r) y = 0, where |α3| + |β2|≠ 0, and {αi}3i=0, {βi}2i=0, {εi}1i=0 are real parameters. The existence conditions for the polynomial solutions are given. A simple procedure based on a recurrence relation is introduced to evaluate these polynomial solutions explicitly. For α0 = 0, α1≠ 0, we prove that the polynomial solutions of the corresponding differential equation are sources of finite sequences of orthogonal polynomials. Several mathematical properties, such as the recurrence relation, Christoffel-Darboux formulas and the norms of these polynomials, are discussed. We shall also show that they exhibit a factorization property that permits the construction of other infinite sequences of orthogonal polynomials.https://ojs.cvut.cz/ojs/index.php/ap/article/view/7583heun equationconfluent forms of heun’s equationpolynomial solutionssequences of orthogonal polynomials |
| spellingShingle | Nasser Saad On generalized Heun equation with some mathematical properties Acta Polytechnica heun equation confluent forms of heun’s equation polynomial solutions sequences of orthogonal polynomials |
| title | On generalized Heun equation with some mathematical properties |
| title_full | On generalized Heun equation with some mathematical properties |
| title_fullStr | On generalized Heun equation with some mathematical properties |
| title_full_unstemmed | On generalized Heun equation with some mathematical properties |
| title_short | On generalized Heun equation with some mathematical properties |
| title_sort | on generalized heun equation with some mathematical properties |
| topic | heun equation confluent forms of heun’s equation polynomial solutions sequences of orthogonal polynomials |
| url | https://ojs.cvut.cz/ojs/index.php/ap/article/view/7583 |
| work_keys_str_mv | AT nassersaad ongeneralizedheunequationwithsomemathematicalproperties |