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Ratios of Entire Functions and Generalized Stieltjes Functions

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Abstract

Monotonicity properties of the ratio

$$\begin{aligned} \log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, \end{aligned}$$

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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Authors and Affiliations

  1. Department of Psychology, University of Copenhagen, Øster Farimagsgade 2A, DK-1353, Copenhagen, Denmark

    Dimitris Askitis

  2. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100, Copenhagen, Denmark

    Henrik L. Pedersen

Authors
  1. Dimitris Askitis
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  2. Henrik L. Pedersen
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Correspondence to Henrik L. Pedersen.

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Communicated by Elias Wegert.

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Askitis, D., Pedersen, H.L. Ratios of Entire Functions and Generalized Stieltjes Functions. Comput. Methods Funct. Theory 22, 471–489 (2022). https://doi.org/10.1007/s40315-021-00405-5

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  • DOI: https://doi.org/10.1007/s40315-021-00405-5

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