Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable
We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials {Pₙ}ₙ₌₀ ∞, the orthogonality of the second derivatives {²ₓPₙ}ₙ₌₂ ∞ , and a generalized Sturm-Liouville type equation. Our treatment of the generalized Bochner theorem...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2018 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2018
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/209878 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable / M. Kenfack Nangho, K. Jordaan // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862634882951282688 |
|---|---|
| author | Kenfack Nangho, M. Jordaan, K. |
| author_facet | Kenfack Nangho, M. Jordaan, K. |
| citation_txt | Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable / M. Kenfack Nangho, K. Jordaan // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials {Pₙ}ₙ₌₀ ∞, the orthogonality of the second derivatives {²ₓPₙ}ₙ₌₂ ∞ , and a generalized Sturm-Liouville type equation. Our treatment of the generalized Bochner theorem leads to explicit solutions of the difference equation [Vinet L., Zhedanov A., J. Comput. Appl. Math. 211 (2008), 45-56], which proves that the only monic orthogonal polynomials that satisfy the first structure relation are Wilson polynomials, continuous dual Hahn polynomials, Askey-Wilson polynomials, and their special or limiting cases as one or more parameters tend to ∞. This work extends our previous result [arXiv:1711.03349] concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying the first structure relation.
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| first_indexed | 2025-12-07T13:58:39Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-209878 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T13:58:39Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kenfack Nangho, M. Jordaan, K. 2025-11-28T09:39:20Z 2018 Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable / M. Kenfack Nangho, K. Jordaan // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D45; 33C45; 42C05 arXiv: 1801.10554 https://nasplib.isofts.kiev.ua/handle/123456789/209878 https://doi.org/10.3842/SIGMA.2018.126 We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials {Pₙ}ₙ₌₀ ∞, the orthogonality of the second derivatives {²ₓPₙ}ₙ₌₂ ∞ , and a generalized Sturm-Liouville type equation. Our treatment of the generalized Bochner theorem leads to explicit solutions of the difference equation [Vinet L., Zhedanov A., J. Comput. Appl. Math. 211 (2008), 45-56], which proves that the only monic orthogonal polynomials that satisfy the first structure relation are Wilson polynomials, continuous dual Hahn polynomials, Askey-Wilson polynomials, and their special or limiting cases as one or more parameters tend to ∞. This work extends our previous result [arXiv:1711.03349] concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying the first structure relation. The research of MKN was supported by a Vice-Chancellor’s Postdoctoral Fellowship from the University of Pretoria. The research by KJ was partially supported by the National Research Foundation of South Africa under grant number 108763. MKN thanks the African Institute for Mathematical Sciences, Muizenberg, South Africa, for their hospitality during his research visit in January 2018, where this paper was completed. We thank the referees for their careful consideration of the manuscript and helpful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable Article published earlier |
| spellingShingle | Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable Kenfack Nangho, M. Jordaan, K. |
| title | Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable |
| title_full | Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable |
| title_fullStr | Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable |
| title_full_unstemmed | Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable |
| title_short | Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable |
| title_sort | structure relations of classical orthogonal polynomials in the quadratic and q-quadratic variable |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209878 |
| work_keys_str_mv | AT kenfacknanghom structurerelationsofclassicalorthogonalpolynomialsinthequadraticandqquadraticvariable AT jordaank structurerelationsofclassicalorthogonalpolynomialsinthequadraticandqquadraticvariable |