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Hypergeometric functions and irrationality measures. (English) Zbl 0904.11020
Motohashi, Y. (ed.), Analytic number theory. Proceedings of the 39th Taniguchi international symposium on mathematics, Kyoto, Japan, May 13–17, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 247, 353-360 (1997).
The author shows how the analogue for one-dimensional Euler-Pochhammer integrals of the method developed in [G. Rhin and C. Viola, Acta Arith. 77, No. 1, 23-56 (1996; Zbl 0864.11037)] can be applied to obtain very easily good irrationality measures for the value of the logarithm at rational points. Let $$h, j, l$$ be integers satisfying $$h>\max\{0,-l\}$$, $$j>\max\{0,l\}$$, and define $$M=\max\{j-l,h+l\}$$. By applying the Euler-Pochhammer integral representation $_2F_1(\alpha,\beta;\gamma;y)= {\Gamma(\gamma)\over\Gamma(\beta)\Gamma(\gamma-\beta)}\int_0^1 {x^{\beta-1}(1-x)^{\gamma-\beta-1}\over(1-xy)^\alpha}dx$ of the Gauss hypergeometric function, one obtains the least irrationality measure $\mu(\log(1+r/s))\leq{U\over V}$ for integers $$r$$ and $$s$$ satisfying $$r\neq 0$$, $$s\geq 1$$, $$r>-s$$, $$(r,s)=1$$, provided that $$V>0$$. Here, $U=\log| f(x_1)| -\log f(x_0)$ and $V=-\log f(x_0)+\int_{\Omega}d\psi(x)-M(1+\log s)-(h+j)\log| r/s| +\min\{0,l\}\log(1+r/s),$ where $$\Omega$$ is the set of $$\omega\in[0,1)$$ satisfying $$[(j-l)\omega]+[(h+l)\omega]<[h\omega]+[j\omega]$$, $$\psi(x)=\Gamma'(x)/\Gamma(x)$$, $$x_0$$ and $$x_1$$ are the stationary points $$\neq 0,1$$ of the function $f(x)={x^h(1-x)^j\over(1+(r/s)x)^{j-l}}$ with $$0<x_0<1$$ and $$1+(r/s)x_1<0$$. The method yields the best known irrationality measures of a class of logarithms of rational numbers. Especially, a simple proof of the best known irrationality measure of $$\log 2$$, $$3.89139978$$ [E. A. Rukhadze, Mosc. Univ. Math. Bull. 42, No. 6, 30-35 (1987); translation from Vestn. Mosk. Univ., Ser. I 1987, No. 6, 25-29 (1987; Zbl 0635.10025)], can be obtained.
For the entire collection see [Zbl 0874.00035].

##### MSC:
 11J91 Transcendence theory of other special functions 11J82 Measures of irrationality and of transcendence 33C05 Classical hypergeometric functions, $${}_2F_1$$