  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 $\mathcal A_k$.
