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


SIGMA 13 (2017), 092, 18 pages      arXiv:1706.10087      https://doi.org/10.3842/SIGMA.2017.092

A Variation of the $q$-Painlevé System with Affine Weyl Group Symmetry of Type $E_7^{(1)}$

Hidehito Nagao
Department of Arts and Science, National Institute of Technology, Akashi College, Hyogo 674-8501, Japan

Received July 03, 2017, in final form November 24, 2017; Published online December 10, 2017

Abstract
Recently a certain $q$-Painlevé type system has been obtained from a reduction of the $q$-Garnier system. In this paper it is shown that the $q$-Painlevé type system is associated with another realization of the affine Weyl group symmetry of type $E_7^{(1)}$ and is different from the well-known $q$-Painlevé system of type $E_7^{(1)}$ from the point of view of evolution directions. We also study a connection between the $q$-Painlevé type system and the $q$-Painlevé system of type $E_7^{(1)}$. Furthermore determinant formulas of particular solutions for the $q$-Painlevé type system are constructed in terms of the terminating $q$-hypergeometric function.

Key words: $q$-Painlevé system of type $E_7^{(1)}$; $q$-Garnier system; Padé method; $q$-hypergeometric function.

pdf (518 kb)   tex (24 kb)

References

  1. Dzhamay A., Sakai H., Takenawa T., Discrete Schlesinger transformations, their Hamiltonian formulation, and difference Painlevé equations, arXiv:1302.2972.
  2. Dzhamay A., Takenawa T., Geometric analysis of reductions from Schlesinger transformations to difference Painlevé equations, in Algebraic and Analytic Aspects of Integrable Systems and Painlevé Equations, Contemp. Math., Vol. 651, Amer. Math. Soc., Providence, RI, 2015, 87-124, arXiv:1408.3778.
  3. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
  4. Grammaticos B., Ohta Y., Ramani A., Sakai H., Degeneration through coalescence of the $q$-Painlevé VI equation, J. Phys. A: Math. Gen. 31 (1998), 3545-3558.
  5. Grammaticos B., Ramani A., On a novel $q$-discrete analogue of the Painlevé VI equation, Phys. Lett. A 257 (1999), 288-292.
  6. Ikawa Y., Hypergeometric solutions for the $q$-Painlevé equation of type $E^{(1)}_6$ by the Padé method, Lett. Math. Phys. 103 (2013), 743-763, arXiv:1207.6446.
  7. Jacobi C.G.J., Über die Darstellung einer Reihe gegebner Werthe durch eine gebrochne rationale Function, J. Reine Angew. Math. 30 (1846), 127-156.
  8. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau $-function, Phys. D 2 (1981), 306-352.
  9. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
  10. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III, Phys. D 4 (1981), 26-46.
  11. Jimbo M., Sakai H., A $q$-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145-154.
  12. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., ${}_{10}E_9$ solution to the elliptic Painlevé equation, J. Phys. A: Math. Gen. 36 (2003), L263-L272, nlin.SI/0303032.
  13. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., Hypergeometric solutions to the $q$-Painlevé equations, Int. Math. Res. Not. 2004 (2004), 2497-2521, nlin.SI/0403036.
  14. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., Construction of hypergeometric solutions to the $q$-Painlevé equations, Int. Math. Res. Not. 2005 (2005), 1441-1463, nlin.SI/0501051.
  15. Kajiwara K., Noumi M., Yamada Y., $q$-Painlevé systems arising from $q$-KP hierarchy, Lett. Math. Phys. 62 (2002), 259-268, nlin.SI/0112045.
  16. Kajiwara K., Noumi M., Yamada Y., Geometric aspects of Painlevé equations, J. Phys. A: Math. Theor. 50 (2017), 073001, 164 pages, arXiv:1509.08186.
  17. Knuth D.E., Two notes on notation, Amer. Math. Monthly 99 (1992), 403-422.
  18. Mano T., Determinant formula for solutions of the Garnier system and Padé approximation, J. Phys. A: Math. Theor. 45 (2012), 135206, 14 pages.
  19. Mano T., Tsuda T., Two approximation problems by Hermite and the Schlesinger transformations, in Novel Development of Nonlinear Discrete Integrable Systems, RIMS Kôkyûroku Bessatsu, Vol. B47, Res. Inst. Math. Sci. (RIMS), Kyoto, 2014, 77-86.
  20. Mano T., Tsuda T., Hermite-Padé approximation, isomonodromic deformation and hypergeometric integral, Math. Z. 285 (2017), 397-431, arXiv:1502.06695.
  21. Masuda T., Hypergeometric $\tau$-functions of the $q$-Painlevé system of type $E_7^{(1)}$, SIGMA 5 (2009), 035, 30 pages, arXiv:0903.4102.
  22. Masuda T., Hypergeometric $\tau$-functions of the $q$-Painlevé system of type $E^{(1)}_8$, Ramanujan J. 24 (2011), 1-31.
  23. Murata M., Lax forms of the $q$-Painlevé equations, J. Phys. A: Math. Theor. 42 (2009), 115201, 17 pages, arXiv:0810.0058.
  24. Nagao H., The Padé interpolation method applied to $q$-Painlevé equations, Lett. Math. Phys. 105 (2015), 503-521, arXiv:1409.3932.
  25. Nagao H., The Padé interpolation method applied to $q$-Painlevé equations II (differential grid version), Lett. Math. Phys. 107 (2017), 107-127, arXiv:1509.05892.
  26. Nagao H., Lax pairs for additive difference Painlevé equations, arXiv:1604.02530.
  27. Nagao H., Hypergeometric special solutions for $d$-Painlevé equations, arXiv:1706.10101.
  28. Nagao H., Yamada Y., Study of $q$-Garnier system by Padé method, arXiv:1601.01099.
  29. Nagao H., Yamada Y., Variations of $q$-Garnier system, arXiv:1710.03998.
  30. 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.
  31. 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.
  32. Ohyama Y., Kawamuko H., Sakai H., Okamoto K., Studies on the Painlevé equations. V. Third Painlevé equations of special type $P_{\rm III}(D_7)$ and $P_{\rm III}(D_8)$, J. Math. Sci. Univ. Tokyo 13 (2006), 145-204.
  33. Ohyama Y., Okumura S., A coalescent diagram of the Painlevé equations from the viewpoint of isomonodromic deformations, J. Phys. A: Math. Gen. 39 (2006), 12129-12151, math.CA/0601614.
  34. Okamoto K., Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Japan. J. Math. (N.S.) 5 (1979), 1-79.
  35. Ormerod C.M., Rains E.M., Commutation relations and discrete Garnier systems, SIGMA 12 (2016), 110, 50 pages, arXiv:1601.06179.
  36. Ormerod C.M., Rains E.M., An elliptic Garnier system, Comm. Math. Phys. 355 (2017), 741-766, arXiv:1607.07831.
  37. 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.
  38. Ramani A., Grammaticos B., Tamizhmani T., Tamizhmani K.M., Special function solutions of the discrete Painlevé equations, Comput. Math. Appl. 42 (2001), 603-614.
  39. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
  40. Sakai H., Hypergeometric solution of $q$-Schlesinger system of rank two, Lett. Math. Phys. 73 (2005), 237-247.
  41. Sakai H., A $q$-analog of the Garnier system, Funkcial. Ekvac. 48 (2005), 273-297.
  42. Sakai H., Lax form of the $q$-Painlevé equation associated with the $A^{(1)}_2$ surface, J. Phys. A: Math. Gen. 39 (2006), 12203-12210.
  43. Suzuki T., A $q$-analogue of the Drinfeld-Sokolov hierarchy of type $A$ and $q$-Painlevé system, in Algebraic and Analytic Aspects of Integrable Systems and Painlevé Equations, Contemp. Math., Vol. 651, Amer. Math. Soc., Providence, RI, 2015, 25-38, arXiv:1105.4240.
  44. Suzuki T., A reformulation of the generalized $q$-Painlevé VI system with $W(A^{(1)}_{2n+1})$ symmetry, J. Integrable Syst. 2 (2017), xyw017, 18 pages, arXiv:1602.01573.
  45. Takenawa T., Weyl group symmetry of type $D^{(1)}_5$ in the $q$-Painlevé V equation, Funkcial. Ekvac. 46 (2003), 173-186.
  46. Yamada Y., A Lax formalism for the elliptic difference Painlevé equation, SIGMA 5 (2009), 042, 15 pages, arXiv:0811.1796.
  47. Yamada Y., Padé method to Painlevé equations, Funkcial. Ekvac. 52 (2009), 83-92.
  48. Yamada Y., Lax formalism for $q$-Painlevé equations with affine Weyl group symmetry of type $E^{(1)}_n$, Int. Math. Res. Not. 2011 (2011), 3823-3838, arXiv:1004.1687.
  49. Yamada Y., A simple expression for discrete Painlevé equations, in Novel Development of Nonlinear Discrete Integrable Systems, RIMS Kôkyûroku Bessatsu, Vol. B47, Res. Inst. Math. Sci. (RIMS), Kyoto, 2014, 87-95.
  50. Yamada Y., An elliptic Garnier system from interpolation, SIGMA 13 (2017), 069, 8 pages, arXiv:1706.05155.

Previous article  Next article   Contents of Volume 13 (2017)