This article is concerned with a value distribution of the fifth Painlevé transcendents in sectorial domains around a fixed singular point. We show that the cardinality of the 1-points of a fifth Painlevé transcendent in a sector has an asymptotic growth of finite order, thereby giving an improvement of the known estimates.

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This article concerns the value distribution of the third Painlevé transcendents in sectorial domains around fixed singular points. We show that the cardinality of the zeros of a third Painlevé transcendent in a sector has an asymptotic growth of finite order, thereby giving an improvement of the known estimation.

This article concerns with the second Painlevé hierarchy, i.e., the 2nth order analogues of the second Painlevé equation. Though several higher order analogues of the second Painlevé equation are proposed by several authors, we investigate one derived from the KdV hierarchy by similarity reduction, and give the lower estimate of the growth order of transcendental meromorphic solutions to the second Painlevé hierarchy with an integer parameter. Moreover, the first lemma in Appendix supplements an incompleteness of Li and He [“On analytic properties of higher analogs of the second Painlevé equation,” J. Math. Phys. 43, 1106–1115 (2002)]10.1063/1.1420396 and Li [“Some properties of the solutions of higher analogue of the Painlevé equation,” Acta Math. Appl. Sin. 22, 59–64 (2006)]10.1007/s10255-005-0285-0.

In this article, we study the string equation of type (2,5), which is derived from 2D gravity theory or the string theory. We consider the equation as a 4th order analogue of the first Painlevé equation, take the autonomous limit, and solve it concretely by use of the Weierstrass' elliptic function.