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

Abstract

For $k\in \mathbb{R}$, we consider a $\mathbb{C}$-algebra ${\mathcal{A}}_k$ of holomorphic functions in the half plane $Rez>k$ with (at most) subexponential growth on the real line to $+\infty$. In the ${\mathcal{A}}_k$-algebra of sequences of functions $\{\alpha :\mathbb{N}\to {\mathcal{A}}_k\}$, we consider the ${\mathcal{A}}_k$-subalgebra ${\mathcal{H}}_k$ consisting in those $\alpha$ for which there exists a continuous map $M:\{Rez>k\}\to [0,+\infty )$ such that $|\alpha (n)(z)|\le M(z)n^k$ for all $Rez>k,n\ge 1$, and ${\lim }_{x\to +\infty }{e}^{-ax}M(x)=0$, for all $a>0$. Given $L$ a sequence of holomorphic functions on $Rez>k$ 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 $Rez>k$, then $F_{{\alpha }_1},\dots ,F_{{\alpha }_r}$ are linearly (algebraically) independent over the quotient field of ${\mathcal{A}}_k$.

arXiv comment15 pages; minor corrections; to appear in The Journal of Analysis