Funkcialaj Ekvacioj, 47 (2004) 291-305

New Expressions for Discrete Painleve Equations

Mikio MURATA
The University of Tokyo, Japan

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Abstract. We present a new expression for the elliptic-difference Painleve equation. As in our construction all parameters in their equations appear in a symmetric way, the permutation symmetry of the equations is immediately apparent. We present expressions for other discrete Painleve equations, obtained in a similar way.

Key Words and Phrases. Integrable systems, Difference equations, Painleve-type functions.
2000 Mathematics Subject Classification Numbers. 39A20, 33E17, 14E07.

References

• Grammaticos, B., Ramani, A., and Papageorgiou, V. G., Do integrable mappings have the Painleve property?, Phys. Rev. Lett., 67 (1991), 1825-1828.
• Grammaticos, B., Ohta, Y., Ramani, A., and Sakai, H., Degeneration through coalescence of the $q$-Painleve VI equation, J. Phys. A: Math. Gen., 31 (1998), 3545-3558.
• Jimbo, M. and Sakai, H., A $q$-analog of the sixth Painleve equation, Lett. Math. Phys., 38 (1996), 145-154.
• Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y., and Yamada, Y., ${}_{10}E_9$ solution to the elliptic Painleve equation, J. Phys. A: Math. Gen., 36 (2003), L263-L272.
• Murata, M., Sakai, H., and Yoneda, J., Riccati solutions of discrete Painleve equations with Weyl group symmetry of type $E_8^{(1)}$, J. Math. Phys., 44 (2003), 1396-1414.
• Ohta, Y., Ramani, A., and Grammaticos, B., An affine Weyl group approach to the eight-parameter discrete Painleve equation, J. Phys. A: Math. Gen., 34 (2001), 10523-10532.
• Ramani, A., Grammaticos, B., and Hietarinta, J., Discrete versions of the Painleve equations, Phys. Rev. Lett., 67 (1991), 1829-1832.
• Sakai, H., Rational Surfaces Associated with Affine Root Systems and Geometry of the Painleve Equations, Comm. Math. Phys., 220 (2001), 165-229.