Ratios of Entire Functions and Generalized Stieltjes Functions

Dimitris Askitis; Henrik L. Pedersen

doi:10.1007/s40315-021-00405-5

Ratios of Entire Functions and Generalized Stieltjes Functions

Dimitris Askitis, Henrik L. Pedersen^*

^*Corresponding author for this work

Research output: Contribution to journal › Journal article › Research › peer-review

1 Citation (Scopus)

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Abstract

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 language	English
Journal	Computational Methods and Function Theory
Volume	22
Pages (from-to)	471–489
ISSN	1617-9447
DOIs	https://doi.org/10.1007/s40315-021-00405-5
Publication status	Published - 2022

Bibliographical note

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

Keywords

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

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10.1007/s40315-021-00405-5

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Cite this

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title = "Ratios of Entire Functions and Generalized Stieltjes Functions",

abstract = "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{\textquoteright}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{\textquoteright}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.",

keywords = "Barnes G-function, Entire function, Euler{\textquoteright}s Gamma function, Generalized Stieltjes function, Laplace transform",

author = "Dimitris Askitis and Pedersen, {Henrik L.}",

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year = "2022",

doi = "10.1007/s40315-021-00405-5",

language = "English",

volume = "22",

pages = "471–489",

journal = "Computational Methods and Function Theory",

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TY - JOUR

T1 - Ratios of Entire Functions and Generalized Stieltjes Functions

AU - Askitis, Dimitris

AU - Pedersen, Henrik L.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - Barnes G-function

KW - Entire function

KW - Euler’s Gamma function

KW - Generalized Stieltjes function

KW - Laplace transform

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U2 - 10.1007/s40315-021-00405-5

DO - 10.1007/s40315-021-00405-5

M3 - Journal article

AN - SCOPUS:85112360328

VL - 22

SP - 471

EP - 489

JO - Computational Methods and Function Theory

JF - Computational Methods and Function Theory

SN - 1617-9447

ER -

Ratios of Entire Functions and Generalized Stieltjes Functions

Abstract

Bibliographical note

Keywords

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