The ﬁrst Painlevé hierarchy, which is a sequence of higher order analogues of the ﬁrst Painlevé equation, follows from the singular manifold equations for the mKdV hierarchy. For meromorphic solutions of the ﬁrst Painlevé hierarchy, we give a lower estimate for the number of poles; which is regarded as an extension of one corresponding to the ﬁrst Painlevé equation, and which indicates a conjecture on the growth order. From our main result, two corollaries follow: one is the transcendency of meromorphic solutions, and the other is a lower estimate for the frequency of α-points. An essential part of our proof is estimation of certain sums concerning the poles of each meromorphic solution.
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Shun Shimomura, Poles and α-points of Meromorphic Solutions of the First Painlevé Hierarchy. Publ. Res. Inst. Math. Sci. 40 (2004), no. 2, pp. 471–485DOI 10.2977/PRIMS/1145475811