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On algebraic independence of the values of some E-functions. II. (Russian. English summary) Zbl 0742.11039
Summary: [For Part I, cf. Čas. Pestovani Mat. 115, No. 3, 283-289 (1990; Zbl 0705.11036).]
Let $$\lambda$$, $$\nu$$, $$\mu_ i$$, $$i=1,\ldots,m$$, be rational numbers such that $$\nu\notin\mathbb{Z}$$, $$\mu_ i\notin\mathbb{Z}$$, $$- \lambda\notin\mathbb{Z}^ +$$, $$\nu-\lambda\notin\mathbb{Z}$$, $$\nu-\mu_ i\notin\mathbb{N}$$ (resp. $$\lambda-\nu-\mu_ i\notin\mathbb{N})$$, $$\mu_ i- \lambda\notin\mathbb{Z}^ +$$, $$\mu_ i-\mu_ j\notin\mathbb{Z}\backslash\{0\}$$, $$i,j=1,\ldots,m$$. Let $$A_ 0=H_ 0$$ be the Kummer’s function $A_ 0(z)=H_ 0(z)=A_{\lambda,\nu}(z)=\sum^ \infty_{n=0}{[\nu,n]\over n![\lambda,n]} z^ n$ and let $$A_ k(z)=z^{-\mu_ k}\int^ zt^{\mu_ k-1}A_{k-1}(t)dt$$, (resp. $$H_ k(z)=z^{-\mu_ k}e^ z\int^ zt^{\mu_ k-1}e^{-t}H_{k-1}(t)dt$$), $$k=1,\ldots,m$$, where $$[\alpha,n]=\alpha(\alpha+1)\ldots(\alpha+n-1)$$. Using the well-known fundamental theorem on the algebraic independence of the values of $$E$$- functions, it is proved that for every pair of algebraic numbers $$\xi,\eta\neq 0$$, the numbers $$e^ \eta$$, $$A_ 0(\xi)$$, $$A_ 0'(\xi)$$, $$A_ 1(\xi),\ldots,A_ m(\xi)$$ (resp. $$e^ \eta$$, $$H_ 0(\xi)$$, $$H_ 0'(\xi)$$, $$H_ 1(\xi),\ldots,H_ m(\xi))$$ are algebraically independent.
##### MSC:
 11J91 Transcendence theory of other special functions 11J85 Algebraic independence; Gel’fond’s method
Zbl 0705.11036