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