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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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