The Determinant of an Elliptic Sylvesteresque Matrix
We evaluate the determinant of a matrix whose entries are elliptic hypergeometric terms and whose form is reminiscent of Sylvester matrices. A hypergeometric determinant evaluation of a matrix of this type has appeared in the context of approximation theory, in the work of Feng, Krattenthaler, and X...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209520 |
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| Cite this: | The Determinant of an Elliptic Sylvesteresque Matrix / G. Bhatnagar, C. Krattenthaler // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 13 назв. — англ. |
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Bhatnagar, G. Krattenthaler, C. 2025-11-24T10:27:08Z 2018 The Determinant of an Elliptic Sylvesteresque Matrix / G. Bhatnagar, C. Krattenthaler // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 13 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D67; 15A15 arXiv: 1802.09885 https://nasplib.isofts.kiev.ua/handle/123456789/209520 https://doi.org/10.3842/SIGMA.2018.052 We evaluate the determinant of a matrix whose entries are elliptic hypergeometric terms and whose form is reminiscent of Sylvester matrices. A hypergeometric determinant evaluation of a matrix of this type has appeared in the context of approximation theory, in the work of Feng, Krattenthaler, and Xu. Our determinant evaluation is an elliptic extension of their evaluation, which has two additional parameters (in addition to the base q and nome p found in elliptic hypergeometric terms). We also extend the evaluation to a formula transforming an elliptic determinant into a multiple of another elliptic determinant. This transformation has two further parameters. The proofs of the determinant evaluation and the transformation formula require an elliptic determinant lemma due to Warnaar, and the application of two Cn elliptic formulas that extend Frenkel and Turaev's ₁₀V₉ summation formula and ₁₂V₁₁ transformation formula, results due to Warnaar, Rosengren, Rains, and Coskun and Gustafson. We thank Michael Schlosser for helpful discussions. We also thank the referees for many useful suggestions. Research of the first author was supported by a grant of the Austrian Science Fund (FWF), START grant Y463. Research of the second author was partially supported by the Austrian Science Fund (FWF), grant F50-N15, in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Determinant of an Elliptic Sylvesteresque Matrix Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
The Determinant of an Elliptic Sylvesteresque Matrix |
| spellingShingle |
The Determinant of an Elliptic Sylvesteresque Matrix Bhatnagar, G. Krattenthaler, C. |
| title_short |
The Determinant of an Elliptic Sylvesteresque Matrix |
| title_full |
The Determinant of an Elliptic Sylvesteresque Matrix |
| title_fullStr |
The Determinant of an Elliptic Sylvesteresque Matrix |
| title_full_unstemmed |
The Determinant of an Elliptic Sylvesteresque Matrix |
| title_sort |
determinant of an elliptic sylvesteresque matrix |
| author |
Bhatnagar, G. Krattenthaler, C. |
| author_facet |
Bhatnagar, G. Krattenthaler, C. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We evaluate the determinant of a matrix whose entries are elliptic hypergeometric terms and whose form is reminiscent of Sylvester matrices. A hypergeometric determinant evaluation of a matrix of this type has appeared in the context of approximation theory, in the work of Feng, Krattenthaler, and Xu. Our determinant evaluation is an elliptic extension of their evaluation, which has two additional parameters (in addition to the base q and nome p found in elliptic hypergeometric terms). We also extend the evaluation to a formula transforming an elliptic determinant into a multiple of another elliptic determinant. This transformation has two further parameters. The proofs of the determinant evaluation and the transformation formula require an elliptic determinant lemma due to Warnaar, and the application of two Cn elliptic formulas that extend Frenkel and Turaev's ₁₀V₉ summation formula and ₁₂V₁₁ transformation formula, results due to Warnaar, Rosengren, Rains, and Coskun and Gustafson.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209520 |
| citation_txt |
The Determinant of an Elliptic Sylvesteresque Matrix / G. Bhatnagar, C. Krattenthaler // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 13 назв. — англ. |
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2025-12-07T16:05:49Z |
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