Volume 27, pp. 94-112, 2007.

Pick functions related to entire functions having negative zeros

Henrik L. Pedersen

Abstract

For any sequence $\{a_k\}$ satisfying $0<a_1\le a_2\le \dots$ and $|a_k-k|\le \text{Const}$ we find the Stieltjes representation of the function $$z\mapsto \frac{\log P(z)}{z\mathrm{L}\mathrm{o}\mathrm{g}z},$$ where $P$ denotes the canonical product of genus 1 having $\{-a_k\}$ as its zero set. We also find conditions on the zeros (e.g. $a_k\in [k,k+1]$ for $k\ge 1$) in order that the function $$z\mapsto \frac{-\log P(z)+z\log P(1)}{z\mathrm{L}\mathrm{o}\mathrm{g}z}$$ be a Pick function. We find the corresponding representation in terms of a positive density on the negative axis. We thereby generalize earlier results about the $\mathrm{\Gamma }$-function. We also show that another related function is a Pick function.

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Key words

pick function, canonical product, integral representation

AMS subject classifications

30E20, 30D15, 30E15, 33B15