Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 069, 8 pages      arXiv:1706.05155      https://doi.org/10.3842/SIGMA.2017.069
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

An Elliptic Garnier System from Interpolation

Yasuhiko Yamada
Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

Received June 20, 2017, in final form August 30, 2017; Published online September 02, 2017

Abstract
Considering a certain interpolation problem, we derive a series of elliptic difference isomonodromic systems together with their Lax forms. These systems give a multivariate extension of the elliptic Painlevé equation.

Key words: elliptic difference; isomonodromic systems; Lax form; interpolation problem.

pdf (310 kb)   tex (13 kb)

References

  1. Kajiwara K., Noumi M., Yamada Y., Geometric aspects of Painlevé equations, J. Phys. A: Math. Theor. 50 (2017), 073001, 164 pages, arXiv:1509.08186.
  2. Nagao H., Yamada Y., Study of $q$-Garnier system by Padé method, Funkcial. Ekvac., to appear, arXiv:1601.01099.
  3. Nijhoff F., Delice N., On elliptic Lax pairs and isomonodromic deformation systems for elliptic lattice equations, arXiv:1605.00829.
  4. Noumi M., Tsujimoto S., Yamada Y., Padé interpolation for elliptic Painlevé equation, in Symmetries, Integrable Systems and Representations, Springer Proc. Math. Stat., Vol. 40, Springer, Heidelberg, 2013, 463-482, arXiv:1204.0294.
  5. Ohta Y., Ramani A., Grammaticos B., An affine Weyl group approach to the eight-parameter discrete Painlevé equation, J. Phys. A: Math. Gen. 34 (2001), 10523-10532.
  6. Ormerod C.M., Rains E.M., Commutation relations and discrete Garnier systems, SIGMA 12 (2016), 110, 50 pages, arXiv:1601.06179.
  7. Ormerod C.M., Rains E.M., An elliptic Garnier system, Comm. Math. Phys. 355 (2017), 741-766, arXiv:1607.07831.
  8. Rains E.M., An isomonodromy interpretation of the hypergeometric solution of the elliptic Painlevé equation (and generalizations), SIGMA 7 (2011), 088, 24 pages, arXiv:0807.0258.
  9. Rains E.M., The noncommutative geometry of elliptic difference equations, arXiv:1607.08876.
  10. Ruijsenaars S.N.M., First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), 1069-1146.
  11. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
  12. Spiridonov V.P., Essays on the theory of elliptic hypergeometric functions, Russian Math. Surveys 63 (2008), 405-472, arXiv:0805.3135.
  13. Yamada Y., A Lax formalism for the elliptic difference Painlevé equation, SIGMA 5 (2009), 042, 15 pages, arXiv:0811.1796.
  14. Yamada Y., Padé method to Painlevé equations, Funkcial. Ekvac. 52 (2009), 83-92.
  15. Zhedanov A.S., Padé interpolation table and biorthogonal rational functions, in Elliptic Integrable Systems, Rokko Lectures in Mathematics, Vol. 18, Kobe University, 2005, 323-363.

Previous article  Next article   Contents of Volume 13 (2017)