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A note on the linear independence of a class of series of functions

  • Mircea CimpoeaşEmail author
Original Research Paper
  • 1 Downloads

Abstract

For \(k\in {\mathbb {R}}\), we consider a \({\mathbb {C}}\)-algebra \({\mathcal {A}}_k\) of holomorphic functions in the half plane \(\mathrm{Re}\,z>k\) with (at most) subexponential growth on the real line to \(+\infty \). In the \({\mathcal {A}}_k\)-algebra of sequences of functions \(\{\alpha :{\mathbb {N}}\rightarrow {\mathcal {A}}_k\}\), we consider the \({\mathcal {A}}_k\)-subalgebra \({\mathcal {H}}_k\) consisting in those \(\alpha \) for which there exists a continuos map \(M:\{\mathrm{Re}\,z>k\}\rightarrow [0,+\infty )\) such that \(|\alpha (n)(z)|\le M(z)n^k\) for all \(\mathrm{Re}\,z>k,n\ge 1\), and \(\lim _{x\rightarrow +\infty }e^{-ax}M(x)=0\), for all \(a>0\). Given L a sequence of holomorphic functions on \(\mathrm{Re}\,z>k\) which satisfies certain conditions, we prove that the map \(\alpha \mapsto F_L(\alpha )\), where \(F_L(\alpha ):=\sum _{n=1}^{+\infty }\alpha (n)(z)L(n)(z)\), is an injective morphism of \({\mathcal {A}}_k\)-modules (or \({\mathcal {A}}_k\)-algebras). Consequently, if \(n\mapsto \alpha _j(n)(z)\in {\mathbb {C}}\), \(1\le j\le r\), are linearly (algebraically) independent over \({\mathbb {C}}\), for z in a nondiscrete subset of \(\mathrm{Re}\,z>k\), then \(F_{\alpha _1},\ldots ,F_{\alpha _r}\) are linearly (algebraically) independent over the quotient field of \({\mathcal {A}}_k\).

Keywords

Series of functions Meromorphic functions Dirichlet series 

Mathematics subject classification

30B50 30D30 

Notes

Acknowledgements

I express my gratitude to Florin Nicolae for valuable discussions regarding the results of this paper.

Compliance with ethical standards

Conflict of interest

This article was not funded. The author declares that he has not conflict of interest.

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Copyright information

© Forum D'Analystes, Chennai 2019

Authors and Affiliations

  1. 1.Simion Stoilow Institute of MathematicsBucharestRomania

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