Differential Operators of Infinite Order and the Distribution of Zeros of Entire Functions

whenever the right-hand side of (1.2) represents an analytic function in a neighborhood of the origin. When φ(x) is an entire function, the operator φ(D) has been studied by several authors (see, for example, [5, §11], [19, Chapter IX], [22] and [32]). The conjecture of Pólya and Wiman, proved in [8], [9] and [17], states that if f(x) ∈ L-P∗, then Df(x) is in the Laguerre-Pólya class for all sufficiently large positive integers m. In this paper, we analyze the more general situation when D is replaced by the operator φ(D). In Section 2, we consider real power series with zero linear term (i.e., α1 = 0 in (1.1)) and α0α2 < 0, and show that if f(x) is any real polynomial, then [φ(D)]f(x) ∈ L-P for all sufficiently large positive integers m (Theorem 2.4). If the linear term in (1.1) is nonzero,