On a characteristic property of periodic entire functions

Introduction.We shall pursue an investigation on a certain functional equation treated in [8] with some overmuch restrictions.The functional equation is related to the following problem: If two entire functions in a certain class have the same zero-sets (including multiplicities), then what can be said about these functions ?Denoting by Gφ) the class of all entire functions each of which is periodic with period b (ΦO) mod a non-constant entire function of order less than one (cf.Def. in § 1), in this paper, we shall prove that if two entire functions belonging to Gφj) 0-1, 2) have the same zero-sets (essentially), then they must coincide up to a non-zero multiplicative constant (Theorem 1).In the proof, we use the Borel-Nevanlinna type unicity theorem.Note that, together with Gφ), the class Jφ), consisting of the entire functions each of which is periodic mod a non-constant polynomial of degree one, is significant in factorization theory (under composition) of transcendental entire functions (cf.for example, [1] or [7]).Now recall some of the results of Gross ([2]).Among others, he proved that any non-constant, periodic, entire function H(z) has an infinite number of fixed points, that is, the zeros of H{z)-z.Further the fixed points play an important role especially in cases concerning periodic entire functions (in factorization theory, etc.).So one might expect that periodic entire functions would be uniquely determined by the sets of their fixed points.In this paper, we'll show that this is the case (Theorem 2).Here, the first author wishes to express his gratitude to Professors Y. Kusunoki (Kyoto Univ.) and M. Ozawa (Tokyo Inst. of Technology) for their encouragement and suggestions. Statement of results.At first, we'll give the definition of the class Gφ) explicitly.