A note on the linear independence of a class of series of functions
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866910427389100032 |
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| author | Cimpoeas, Mircea |
| author_facet | Cimpoeas, Mircea |
| contents | 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 $\mathcal A_k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1802_03377 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | A note on the linear independence of a class of series of functions Cimpoeas, Mircea Number Theory Complex Variables 30B50, 30D30, 13G05 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 $\mathcal A_k$. |
| title | A note on the linear independence of a class of series of functions |
| topic | Number Theory Complex Variables 30B50, 30D30, 13G05 |
| url | https://arxiv.org/abs/1802.03377 |