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On the irrationality of values of hypergeometric functions. (Russian) Zbl 0575.10029
Denote the hypergeometric function by \[ F(\alpha,\beta,\gamma;x)=\sum^{\infty}_{n=0}\frac{[\alpha,n][\beta,n]} {[\gamma,n] n!}x^ n. \] For \(s\geq 1\), put \(\Phi_ j(x)=F(\alpha_ j,1,\alpha_ j+\alpha_ 0;x)\), \(j=1,...,s\) and \(\eta_ j=\Phi_ j(\xi)\), \(j=1,...,s\). In the present paper the following theorems are proved.
Theorem 1. Let \(\alpha_ j=p_ j/q_ j\) with \(p_ j>0\), \(q_ j>0\) in \({\mathbb{Z}}\), \((p_ j,q_ j)=1\), \(j=0,1,...,s\) and \(\alpha_ i-\alpha_ j\not\in {\mathbb{Z}}\), \(1\leq i<j\leq s\). Let \(\xi =a/b\), where a,b are non- zero integers in an imaginary quadratic field I. Suppose further that \(| b| >| a|^{s+1} \exp \{3s(s+2)q_ 0 \bar q\},\) where \(\bar q=\max (q_ 1,...,q_ s).\) Then \(1,\eta_ 1,...,\eta_ s\) are linearly independent over I and there exists an effective constant \(C>0\) such that the inequality \[ | x_ 0+x_ 1\eta_ 1+...+x_ s\eta_ s| \geq C/X^{\kappa} \] holds for all integers \(x_ 0,x_ 1,...,x_ s\) in I, where \(X=\max (| x_ 1|,...,| x_ s|)\) and \[ \kappa =\frac{\log | b| +(3s+11)q_ o \bar q}{s^{-1} \log | b/a| -\log | a| -(3s+6)q_ 0 \bar q}. \] Theorem 2. Suppose that \(\alpha_ 0,\alpha_ 1,...,\alpha_ s\) are as in Theorem 1; \(\xi =a/b\) with a,b being non-zero integers in an imaginary quadratic field I; \(\epsilon >0\). Then there exists a constant \(\gamma\geq 1\) such that if a,b satisfy the condition \(\gamma | a|^ 2<| b|^{\epsilon /(s+1+s\epsilon)},\) then the inequality \[ | x_ 0+x_ 1\eta_ 1+...+x_ s\eta_ s| >| x_ 1...x_ s|^{-1-\epsilon} \] holds for all integers \(x_ 0,x_ 1,...,x_ s\) in I satisfying \(| x_ 1...x_ s| \geq \Xi_ 0\) for some effective constant \(\Xi_ 0>0\).
Reviewer: Kunrui Yu

MSC:
11J81 Transcendence (general theory)
11J17 Approximation by numbers from a fixed field
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