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

TL;DR

This paper investigates the linear independence of certain series of holomorphic functions within a specific algebraic framework, establishing conditions under which linear independence is preserved through a summation map.

Contribution

It introduces a new algebraic structure for series of functions and proves the injectivity of a summation map, linking linear independence in sequences to that of their series.

Findings

The summation map is an injective morphism under certain conditions.

Linear independence of sequences implies independence of their series.

Provides a framework for analyzing series of functions in complex half-planes.

Abstract

For $k\in\mathbb R$, we consider a $\mathbb C$-algebra $\mathcal A_k$ of holomorphic functions in the half plane $Re\; z>k$ 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 continuous map $M:\{Re\; z>k\}\rightarrow [0,+\infty)$ such that $|\alpha(n)(z)|\leq M(z)n^k$ for all $Re\; z>k,n\geq 1$, and $\lim_{x\rightarrow +\infty}e^{-ax}M(x)=0$, for all $a>0$. Given $L$ a sequence of holomorphic functions on $Re\; z>k$ 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 $n\mapsto \alpha_j(n)(z)\in\mathbb C$, $1\leq j\leq r$, are linearly (algebraically) independent over $\mathbb C$, for $z$ in a nondiscrete subset of $Re\; z>k$, then $F_{\alpha_1},\ldots,F_{\alpha_r}$ are linearly (algebraically) independent over the quotient field of $\mathcal A_k$.