On Certain Wronskians of Multiple Orthogonal Polynomials
We consider determinants of Wronskian type whose entries are multiple orthogonal polynomials associated with a path connecting two multi-indices. By assuming that the weight functions form an algebraic Chebyshev (AT) system, we show that the polynomials represented by the Wronskians keep a constant...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2014 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2014
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/146536 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On Certain Wronskians of Multiple Orthogonal Polynomials/ L. Zhang, G. Filipuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 60 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862715652230348800 |
|---|---|
| author | Zhang, L. Filipuk, G. |
| author_facet | Zhang, L. Filipuk, G. |
| citation_txt | On Certain Wronskians of Multiple Orthogonal Polynomials/ L. Zhang, G. Filipuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 60 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We consider determinants of Wronskian type whose entries are multiple orthogonal polynomials associated with a path connecting two multi-indices. By assuming that the weight functions form an algebraic Chebyshev (AT) system, we show that the polynomials represented by the Wronskians keep a constant sign in some cases, while in some other cases oscillatory behavior appears, which generalizes classical results for orthogonal polynomials due to Karlin and Szegő. There are two applications of our results. The first application arises from the observation that the m-th moment of the average characteristic polynomials for multiple orthogonal polynomial ensembles can be expressed as a Wronskian of the type II multiple orthogonal polynomials. Hence, it is straightforward to obtain the distinct behavior of the moments for odd and even m in a special multiple orthogonal ensemble - the AT ensemble. As the second application, we derive some Turán type inequalities for multiple Hermite and multiple Laguerre polynomials (of two kinds). Finally, we study numerically the geometric configuration of zeros for the Wronskians of these multiple orthogonal polynomials. We observe that the zeros have regular configurations in the complex plane, which might be of independent interest.
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| first_indexed | 2025-12-07T17:59:37Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146536 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:59:37Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Zhang, L. Filipuk, G. 2019-02-09T20:41:46Z 2019-02-09T20:41:46Z 2014 On Certain Wronskians of Multiple Orthogonal Polynomials/ L. Zhang, G. Filipuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 60 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 05E35; 11C20; 12D10; 26D05; 41A50 DOI:10.3842/SIGMA.2014.103 https://nasplib.isofts.kiev.ua/handle/123456789/146536 We consider determinants of Wronskian type whose entries are multiple orthogonal polynomials associated with a path connecting two multi-indices. By assuming that the weight functions form an algebraic Chebyshev (AT) system, we show that the polynomials represented by the Wronskians keep a constant sign in some cases, while in some other cases oscillatory behavior appears, which generalizes classical results for orthogonal polynomials due to Karlin and Szegő. There are two applications of our results. The first application arises from the observation that the m-th moment of the average characteristic polynomials for multiple orthogonal polynomial ensembles can be expressed as a Wronskian of the type II multiple orthogonal polynomials. Hence, it is straightforward to obtain the distinct behavior of the moments for odd and even m in a special multiple orthogonal ensemble - the AT ensemble. As the second application, we derive some Turán type inequalities for multiple Hermite and multiple Laguerre polynomials (of two kinds). Finally, we study numerically the geometric configuration of zeros for the Wronskians of these multiple orthogonal polynomials. We observe that the zeros have regular configurations in the complex plane, which might be of independent interest. We thank the referees for helpful comments, suggestions, and pointing out the additional references
 [23, 24, 44, 46]. LZ is partially supported by The Program for Professor of Special
 Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (No. SHH1411007)
 and by Grant SGST 12DZ 2272800 from Fudan University. GF is supported by the MNiSzW
 Iuventus Plus grant Nr 0124/IP3/2011/71. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Certain Wronskians of Multiple Orthogonal Polynomials Article published earlier |
| spellingShingle | On Certain Wronskians of Multiple Orthogonal Polynomials Zhang, L. Filipuk, G. |
| title | On Certain Wronskians of Multiple Orthogonal Polynomials |
| title_full | On Certain Wronskians of Multiple Orthogonal Polynomials |
| title_fullStr | On Certain Wronskians of Multiple Orthogonal Polynomials |
| title_full_unstemmed | On Certain Wronskians of Multiple Orthogonal Polynomials |
| title_short | On Certain Wronskians of Multiple Orthogonal Polynomials |
| title_sort | on certain wronskians of multiple orthogonal polynomials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146536 |
| work_keys_str_mv | AT zhangl oncertainwronskiansofmultipleorthogonalpolynomials AT filipukg oncertainwronskiansofmultipleorthogonalpolynomials |