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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.14107 |
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| _version_ | 1866929460546109440 |
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| author | Little, Alex |
| author_facet | Little, Alex |
| contents | We present a representation of skew-orthogonal polynomials of symplectic type ($β=4$) in terms of a matrix Riemann-Hilbert problem, for weights of the form ${\rm e}^{-V(z)}$ where $V$ is a polynomial of even degree and positive leading coefficient. This is done by representing skew-orthogonality as a kind of multiple-orthogonality. From this, we derive a $β=4$ analogue of the Christoffel-Darboux formula. Finally, our Riemann-Hilbert representation allows us to derive a Lax pair whose compatibility condition may be viewed as a $β=4$ analogue of the Toda lattice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_14107 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type Little, Alex Mathematical Physics 35Q15 (Primary), 15A52 (Secondary) We present a representation of skew-orthogonal polynomials of symplectic type ($β=4$) in terms of a matrix Riemann-Hilbert problem, for weights of the form ${\rm e}^{-V(z)}$ where $V$ is a polynomial of even degree and positive leading coefficient. This is done by representing skew-orthogonality as a kind of multiple-orthogonality. From this, we derive a $β=4$ analogue of the Christoffel-Darboux formula. Finally, our Riemann-Hilbert representation allows us to derive a Lax pair whose compatibility condition may be viewed as a $β=4$ analogue of the Toda lattice. |
| title | A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type |
| topic | Mathematical Physics 35Q15 (Primary), 15A52 (Secondary) |
| url | https://arxiv.org/abs/2306.14107 |