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The first Painleve hierarchy, which is a sequence of higher order ana- logues of the first Painleve equation, follows from the singular manifold equations for the mKdV hierarchy. For meromorphic solutions of the first Painleve hierarchy, we give a lower estimate for the number of poles; which is regarded as an extension of one corresponding to the first Painleve equa- tion, and which indicates a conjecture on the growth order. From our main result, two corollaries follow: one is the transcendency of meromorphic so- lutions, and the other is a lower estimate for the frequency of fi-points. An essential part of our proof is estimation of certain sums concerning the poles of each meromorphic solution.

Poles and $\alpha$-points of meromorphic solutions of the first Painlevé hierarchy