Ratios of Entire Functions and Generalized Stieltjes Functions

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Monotonicity properties of the ratio logf(x+a1)⋯f(x+an)f(x+b1)⋯f(x+bn),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.

Original languageEnglish
JournalComputational Methods and Function Theory
Volume22
Pages (from-to)471–489
ISSN1617-9447
DOIs
Publication statusPublished - 2022

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

    Research areas

  • Barnes G-function, Entire function, Euler’s Gamma function, Generalized Stieltjes function, Laplace transform

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