On a general method for deriving a fourth-order differential equation satisfied by Laguerre-Hahn orthogonal polynomials with new results for the class 0 analogous to Hermite
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914592528007168 |
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| author | Khalfallah, Mohamed Maroni, Pascal da Rocha, Zélia |
| author_facet | Khalfallah, Mohamed Maroni, Pascal da Rocha, Zélia |
| contents | In this work, we develop a constructive method for deriving four structure relations and a fourth-order linear differential equation satisfied by Laguerre-Hahn orthogonal polynomial sequences. The method relies on a combination of structure relations, their successive derivatives, and algebraic elimination techniques. Particular attention is given to semiclassical and classical families, which are recovered as special cases within this general framework. The approach is systematized in the form of an algorithm. Using symbolic computations, we obtain explicit new results for Laguerre-Hahn polynomials of class zero, analogous to the Hermite case. In addition, we present results for a semiclassical example of class 1. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_23720 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a general method for deriving a fourth-order differential equation satisfied by Laguerre-Hahn orthogonal polynomials with new results for the class 0 analogous to Hermite Khalfallah, Mohamed Maroni, Pascal da Rocha, Zélia Classical Analysis and ODEs 34, 33C45, 33D45, 42C05, 33F10, 68W30, 62-09, 33F05, 65D20, 68-04 In this work, we develop a constructive method for deriving four structure relations and a fourth-order linear differential equation satisfied by Laguerre-Hahn orthogonal polynomial sequences. The method relies on a combination of structure relations, their successive derivatives, and algebraic elimination techniques. Particular attention is given to semiclassical and classical families, which are recovered as special cases within this general framework. The approach is systematized in the form of an algorithm. Using symbolic computations, we obtain explicit new results for Laguerre-Hahn polynomials of class zero, analogous to the Hermite case. In addition, we present results for a semiclassical example of class 1. |
| title | On a general method for deriving a fourth-order differential equation satisfied by Laguerre-Hahn orthogonal polynomials with new results for the class 0 analogous to Hermite |
| topic | Classical Analysis and ODEs 34, 33C45, 33D45, 42C05, 33F10, 68W30, 62-09, 33F05, 65D20, 68-04 |
| url | https://arxiv.org/abs/2605.23720 |