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].


11J91 Transcendence theory of other special functions
11J82 Measures of irrationality and of transcendence
33C05 Classical hypergeometric functions, \({}_2F_1\)