On irrationality measure

M. G. Bashmakova ; E. S. Zolotukhina

Čebyševskij sbornik, Tome 18 (2017) no. 1, pp. 29-43.

Résumé

In the present paper we will consider the generalization of some methods for evaluation of irrationality measures for $\gamma_d=\sqrt{d}\ln{\frac{\sqrt{d} +1}{\sqrt{d}-1}}$ and currently known results overview. The extent of irrationality for various values of Gauss hypergeometric function were estimated repeatedly, in particular for $2F(1,\frac{1}{2},\frac{3}{2};\frac{1}{d})=\sqrt{d}\ln{\frac{\sqrt{d}+1}{\sqrt{d}-1}}$. The first such estimates in some special cases were obtained by D. Rhinn [1], M. Huttner [2], D. Dubitskas [3]. Afterward by K. Vaananen, A. Heimonen and D. Matala-Aho [4] was elaborated the general method, which one made it possible to get upper bounds for irrationality measures of the Gauss hypergeometric function values $F(1,\frac{1}{k},1+\frac{1}{k};\frac{r}{s}), k\in\mathbb N, k\ge 2, \frac{r}{s}\in\mathbb Q, (r,s)=1, \frac{r}{s}\in (-1,1)$. This method used the Jacobi type polynomials to construct rational approach to the hypergeometric function. In [4] have been obtained many certain estimates, and some of them have not been improved till now. But for the special classes of the values of hypergeometric function later were elaborated especial methods, which allowed to get better evaluations. In the papers [5], [6] authors, worked under supervision of V. Kh. Salikhov, obtained better estimates for the extent of irrationality for some specific values $\gamma_d$. In the basis of proofs for that results were lying symmetrized integral constructions. It should be remarked, that lately symmetrized integrals uses very broadly for researching of irrationality measures. By using such integrals were obtained new estimates for $\ln 2$ ([7]), $\ln 3$, $\ln \pi $, ([8], [9]) and other values. Here we present research and compare some of such symmetrized constructions, which earlier allowed to improve upper bounds of irrationality measure for specific values of $\gamma_d$. This work is devoted to the seventieth Doctor of Physical and Mathematical Sciences, Professor Vasily Ivanovich Bernik. In her curriculum vitae, a brief analysis of his scientific work and educational and organizational activities. The work included a list of 80 major scientific works of V. I. Bernik. Bibliography: 17 titles.

Détail

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

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