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Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System
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- Відділення математики
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2012, том 8
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Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System
Інший ID:
2010 Mathematics Subject Classification: 39A05; 42C05; 34M55; 34M56; 33C45; 37K35
DOI: http://dx.doi.org/10.3842/SIGMA.2012.097
DOI: http://dx.doi.org/10.3842/SIGMA.2012.097
Посилання:
Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System / N.S. Witte, C.M. Ormerod // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 18 назв. — англ.
Підтримка:
This research has been supported by the Australian Research Council’s Centre of Excellence for Mathematics and Statistics of Complex Systems. We are grateful for the clarifications by Kenji Kajiwara of results given in [6] and [5] and the assistance of Yasuhiko Yamada in explaining the results of his work [18]. We also appreciate the assistance of Jason Whyte in the preparation of this manuscript.
Дата:
2012
Переглядів:
664
Завантажень:
268
Анотація:
We construct a Lax pair for the E₆⁽¹⁾ q-Painlevé system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices - the q-linear lattice - through a natural generalisation of the big q-Jacobi weight. As a by-product of our construction we derive the coupled first-order q-difference equations for the E₆⁽¹⁾ q-Painlevé system, thus verifying our identification. Finally we establish the correspondences of our result with the Lax pairs given earlier and separately by Sakai and Yamada, through explicit transformations.
