Ratios of Entire functions and generalized Stieltjes functions

TL;DR

This paper explores the monotonicity of ratios involving entire functions, generalizes known results for gamma functions to higher genus functions, and connects these findings to generalized Stieltjes functions and the Prouhet-Tarry-Escott problem.

Contribution

It extends monotonicity properties to entire functions of higher genus with negative zeros and relates derivatives of these ratios to generalized Stieltjes functions, broadening previous results.

Findings

Generalization of monotonicity properties to genus p entire functions.

Relation of derivatives to generalized Stieltjes functions of order p+1.

Application to Barnes multiple gamma functions and connection to Prouhet-Tarry-Escott problem.

Abstract

Monotonicity properties of the ratio $$ \log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, $$ where $f$ is an entire function are investigated. Earlier results for Euler's gamma function and other entire functions of genus 1 are generalised to entire functions of genus $p$ with negative zeros. Derivatives of order comparable to $p$ of the expression above are related to generalised Stieltjes functions of order $p+1$. Our results are applied to the Barnes multiple gamma functions. We also show how recent results on the behaviour of Euler's gamma function on vertical lines can be sharpened and generalised to functions of higher genus. Finally a connection to the so-called Prouhet-Tarry-Escott problem is described.