On algebraic independence of the values of some E-functions. (Russian. English summary) Zbl 0705.11036

Čížek, Jiří

Čas. Pěstování Mat. 115, No. 3, 283-289 (1990).

Summary: Let \(\lambda\), \(\nu\), \(\mu_ i\), \(i=1,...,m\), be rational numbers such that \(\nu\not\in {\mathbb{Z}}\), \(\mu_ i\not\in {\mathbb{Z}}\), \(-\lambda \not\in {\mathbb{Z}}^+,\quad \nu -\lambda \not\in {\mathbb{Z}},\quad \nu -\mu_ i\not\in {\mathbb{N}},\quad \mu_ i-\lambda \not\in {\mathbb{Z}}^+,\quad \mu_ i-\mu_ j\not\in {\mathbb{Z}}\setminus \{0\},\) \(i,j=1,...,m\). Let \(A_ 0\) be the Kummer’s function \[ A_ 0(z)=A_{\lambda,\nu}(z)=\sum^{\infty}_{n=0}\frac{[\nu,n]z^ n}{n![\lambda,n]} \] and let \(A_ k(z)=\sum^{\infty}_{n=0}\frac{[\nu,n]z^ n}{n...(\mu_ k+n)}\), \(k=1,...,m\), where \([\alpha,n]=\alpha (\alpha +1)...(\alpha +n-1).\) It is proved that the functions \(A_ 0,A_ 0',A_ 1,...,A_ m\) are algebraically independent over \({\mathbb{C}}(z)\). By the well-known fundamental theorem on the algebraic independence of the values of E-functions for every algebraic number \(\xi\neq 0\), the numbers \(A_ 0(\xi),A_ 0'(\xi),A_ 1(\xi),...,A_ m(\xi)\) are algebraically independent, too.

MSC:

11J91	Transcendence theory of other special functions
11J85	Algebraic independence; Gel’fond’s method

Keywords:

Kummer’s function; algebraic independence; values of E-functions for every algebraic number