On estimate of irrationality measure of the numbers $\sqrt{4k+3}\ln{\frac{\sqrt{4k+3}+1}{\sqrt{4k+3}-1}}$ and $\frac{1}{\sqrt{k}}\mathrm{arctg}\,{\frac{1}{\sqrt{k}}}$

M. G. Bashmakova ; E. S. Zolotukhina

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 15-29

Résumé

The arithmetic properties of the values of hypergeometric function have been studied by various methods since the paper of C. Siegel in 1929. This direction of the theory of Diophantine approximations was studied by such authors as М. Hata [1]-[2], F. Amoroso and C. Viola [3], A. Heimonen, T. Matala-aho and K. Väänänen [4]-[5] and other. In recent decades, a number of interesting results in this area have been obtained, many of the previously known estimates for the irrationality measures for values of hypergeometric functions, and other variables have been improved. Currently one of the widely used approaches in the construction of estimates of the irrationality measure is the use of integral constructions symmetric with respect to replacement of parameters. Symmetrized integrals have been previously used by different authors, for example in the G. Rhin's article [6], but the most active development of this direction was acquired after the work of V. ,Kh. Salikhov [7], who received a new estimate for $\ln{3}$ using the symmetrized integral. Subsequently, the symmetry of different types allowed to prove a number of significant results. New estimates for some values of the logarithmic function, the function $\mathrm{arctg}\,{x}$, and classical constants were obtained (see, for example, [8] – [18]). In 2014 Q. Wu and L. Wang intensified V. H. Salikhov's result of the irrationality measure of $\ln{3}$ using common symmetrized polynomials $At-B$, where $t=(x-d)^2$ (see [19]). In the V. A. Androsenko's article the idea of symmetry was applied to the integral of Marcovecchio, who previously proved a new estimate for $\ln{2}$ in [21], and it allowed to improve the result for $\pi/3$. This paper is a continuation of article [22] generalizing results for two types of symmetric integral constructions. The first allows to estimate more effectively the measure of irrationality of numbers of the form $\sqrt{d}\ln{\frac{\sqrt{d}+1}{\sqrt{d}-1}}$ at $d=2^{2k+1}, d=4k+1$ for some $k\in\mathbb N$ (see [22]). It is also possible to obtain estimates of the irrationality measure of numbers $\sqrt{4k+3}\ln{\frac{\sqrt{4k+3}+1}{\sqrt{4k+3}-1}},\ k\in\mathbb N$ using this integral. The second considered integral construction makes it possible to estimate the measure of irrationality of some values of the logarithmic function using another type of symmetry, what was discussed in detail in [22]. This integral also allows to estimate the measure of irrationality of values $\frac{1}{\sqrt{k}}\mathrm{arctg}\,{\frac{1}{\sqrt{k}}}$. A generalization of this case is proposed in this paper.

Détail

Keywords: Irrationality measure, Gauss hypergeometric function, symmetrized integrals.