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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
Citations:
Zbl 0705.11036
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