d-Orthogonal Analogs of Classical Orthogonal Polynomials
Classical orthogonal polynomial systems of Jacobi, Hermite, and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner, they are the only systems on the real line with this property....
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2018 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2018
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/209787 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | d-Orthogonal Analogs of Classical Orthogonal Polynomials / E. Horozov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 64 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862721614396784640 |
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| author | Horozov, E. |
| author_facet | Horozov, E. |
| citation_txt | d-Orthogonal Analogs of Classical Orthogonal Polynomials / E. Horozov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 64 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Classical orthogonal polynomial systems of Jacobi, Hermite, and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner, they are the only systems on the real line with this property. Similar results hold for the discrete orthogonal polynomials. In a recent paper, we introduced a natural class of polynomial systems whose members are the eigenfunctions of a differential operator of higher order and which are orthogonal with respect to d measures, rather than one. These polynomial systems enjoy a number of properties that make them a natural analog of the classical orthogonal polynomials. In the present paper, we continue their study. The most important new properties are their hypergeometric representations, which allow us to derive their generating functions and, in some cases, also Mehler-Heine type formulas.
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| first_indexed | 2025-12-07T18:31:29Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-209787 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:31:29Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Horozov, E. 2025-11-26T12:23:59Z 2018 d-Orthogonal Analogs of Classical Orthogonal Polynomials / E. Horozov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 64 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34L20; 30C15; 33E05 arXiv: 1609.06157 https://nasplib.isofts.kiev.ua/handle/123456789/209787 https://doi.org/10.3842/SIGMA.2018.063 Classical orthogonal polynomial systems of Jacobi, Hermite, and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner, they are the only systems on the real line with this property. Similar results hold for the discrete orthogonal polynomials. In a recent paper, we introduced a natural class of polynomial systems whose members are the eigenfunctions of a differential operator of higher order and which are orthogonal with respect to d measures, rather than one. These polynomial systems enjoy a number of properties that make them a natural analog of the classical orthogonal polynomials. In the present paper, we continue their study. The most important new properties are their hypergeometric representations, which allow us to derive their generating functions and, in some cases, also Mehler-Heine type formulas. The author is sincerely grateful to Boris Shapiro for sharing and discussing some polynomial systems studied here. Without this, the current project would probably have never seen the light of day. Also, his advice for the improvement of the text is acknowledged. The author wants to thank the Mathematics Department of Stockholm University for the hospitality in April 2015 and April 2017. Last but not least, the author acknowledges extremely helpful suggestions and corrections made by the referees, which helped to improve the text considerably. This research has been partially supported by the Grant No DN 02-5 of the Bulgarian Fund “Scientific research”. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications d-Orthogonal Analogs of Classical Orthogonal Polynomials Article published earlier |
| spellingShingle | d-Orthogonal Analogs of Classical Orthogonal Polynomials Horozov, E. |
| title | d-Orthogonal Analogs of Classical Orthogonal Polynomials |
| title_full | d-Orthogonal Analogs of Classical Orthogonal Polynomials |
| title_fullStr | d-Orthogonal Analogs of Classical Orthogonal Polynomials |
| title_full_unstemmed | d-Orthogonal Analogs of Classical Orthogonal Polynomials |
| title_short | d-Orthogonal Analogs of Classical Orthogonal Polynomials |
| title_sort | d-orthogonal analogs of classical orthogonal polynomials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209787 |
| work_keys_str_mv | AT horozove dorthogonalanalogsofclassicalorthogonalpolynomials |