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 $Re\; z>k$ with (at most) subexponential growth on the real line to $+\infty$. In the $\mathcal A_k$-algebra of sequences of functions $\{α:\mathbb N\rightarrow \mathcal A_k\}$, we consider the $\mathcal A_k$-subalgebra $\mathcal H_k$ consisting in those $α$ for which there exists a continuous map $M:\{Re\; z>k\}\rightarrow [0,+\infty)$ such that $|α(n)(z)|\leq M(z)n^k$ 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 $α\mapsto F_L(α)$, where $F_L(α):=\sum_{n=1}^{+\infty}α(n)(z)L(n)(z)$, is an injective morphism of $\mathcal A_k$-modules (or $\mathcal A_k$-algebras). Consequently, if $n\mapsto α_j(n)(z)\in\mathbb C$, $1\leq j\leq r$, are linearly (algebraically) independent over $\mathbb C$, for $z$ in a nondiscrete subset of $Re\; z>k$, then $F_{α_1},\ldots,F_{α_r}$ are linearly (algebraically) independent over the quotient field of $\mathcal A_k$.