## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### A certain expression of the first Painlevé hierarchy

Shun Shimomura

#### Abstract

We show that each equation in the first Painlevé hierarchy is equivalent to a system of nonlinear equations determined by a kind of generating function, and that it admits the Painlevé property. Our results are derived from the fact that the first Painlevé hierarchy follows from isomonodromic deformation of certain linear systems with an irregular singular point.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 80, Number 6 (2004), 105-109.

Dates
First available in Project Euclid: 13 May 2005

https://projecteuclid.org/euclid.pja/1116014786

Digital Object Identifier
doi:10.3792/pjaa.80.105

Mathematical Reviews number (MathSciNet)
MR2075451

Zentralblatt MATH identifier
1061.34064

#### Citation

Shimomura, Shun. A certain expression of the first Painlevé hierarchy. Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 6, 105--109. doi:10.3792/pjaa.80.105. https://projecteuclid.org/euclid.pja/1116014786

#### References

• Jimbo, M., Miwa, T., and Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau$-function. Phys. D, 2, 306–352 (1981).
• Kawai, T., Koike, T., Nishikawa, Y., and Takei, Y.: On the Stokes geometry of higher order Painlevé equations. (To appear).
• Kudryashov, N. A.: The first and second Painlevé equations of higher order and some relations between them. Phys. Lett. A, 224, 353–360 (1997).
• Kudryashov, N. A., and Soukharev, M. B.: Uniformization and transcendence of solutions for the first and second Painlevé hierarchies. Phys. Lett. A, 237, 206–216 (1998).
• Miwa, T.: Painlevé property of monodromy preserving deformation equations and the analyticity of $\tau$-functions. Publ. Res. Inst. Math. Sci., 17, 703–721 (1981).
• Shimomura, S.: Painlevé property of a degenerate Garnier system of (9/2)-type and of a certain fourth order non-linear ordinary differential equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29, 1–17 (2000).
• Shimomura, S.: On the Painlevé I hierarchy. RIMS Kokyuroku, 1203, 46–50 (2001).
• Shimomura, S.: Poles and $\alpha$-points of meromorphic solutions of the first Painlevé hierarchy. Publ. Res. Inst. Math. Sci., 40, 471–485 (2004).