Ratios of Entire functions and generalized Stieltjes functions

2021/07/02 by Askitis, Dimitris, Pedersen, Henrik L. · 1 citation
#26A48 (Primary) #30E20 #44A10 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2107.00905

Abstract

Monotonicity properties of the ratio log (f(x+a₁)⋯ f(x+a_n))/(f(x+b₁)⋯ 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.

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