Legendre type polynomials and irrationality measures. (English) Zbl 0692.10034

Hata, Masayoshi

J. Reine Angew. Math. 407, 99-125 (1990).

Diophantine approximations to irrational numbers, such as \(\log 2\), \(\pi/\sqrt{3}\), \(\zeta(2),\) and \(\zeta(3)\) are studied using the following Legendre type polynomials: \[ P_{n,m}(x)=\frac{1}{n!}(x^{n- m}(1-x)^{n+m})^{(n)}. \] The importance of these polynomials is that the greatest common divisor of the coefficients of \(P_{n,m}(x)\) is fairly large. The following measures of irrationality are obtained for large positive integer q: \[ | \log 2-p/q| >q^{-3.8914},\quad | \pi \sqrt{3}-p/q| >q^{-5.0875},\quad | \pi^ 2-p/q| >q^{-7.5252},\quad | \zeta (3)-p/q| >q^{-8.8303}. \]

Reviewer: M.Hata

MSC:

11J81

Transcendence (general theory)

Keywords:

zeta(2); zeta(3); approximations to irrational numbers; Legendre type polynomials; measures of irrationality

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