Geodesic
Approximation of Values of the Gauss Hypergeometric Function by Rational Fractions
Matematičeskie zametki, Tome 88 (2010) no. 6, pp. 822-835 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a new approach to estimating the irrationality measure of numbers that are values of the Gauss hypergeometric function. Some of the previous results are improved, in particular, those concerning irrationalities of the form $\sqrt{k}\ln((\sqrt{k}+1)/(\sqrt{k}-1))$ with $k\in\mathbb N$.
Keywords: Gauss hypergeometric function, irrationality measure, Laplace method
Mots-clés : rational fraction, Laurent series. 
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M. G. Bashmakova. Approximation of Values of the Gauss Hypergeometric Function by Rational Fractions. Matematičeskie zametki, Tome 88 (2010) no. 6, pp. 822-835. http://geodesic.mathdoc.fr/item/MZM_2010_88_6_a2/

[1] A. Heimonen, T. Matala-Aho, K. Väänänen, “An application of Jacobi type polynomials to irrationality measures”, Bull. Austral. Math. Soc., 50:2 (1994), 225–243 | DOI | MR | Zbl

[2] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Spravochnaya matematicheskaya biblioteka, Nauka, M., 1973 | MR | Zbl

[3] G. Rhin, “Approximants de Padé et mesures effectives d'irrationalité”, Séminaire de Théorie des Nombres, Progr. Math., 71, Boston, MA, Birkhäuser Boston, 1987, 155–164 | MR | Zbl

[4] E. S. Salnikova, “O merakh irratsionalnosti nekotorykh znachenii funktsii Gaussa”, Chebyshevskii sb., 8:2 (2007), 88–96 | MR | Zbl

[5] V. Kh. Salikhov, “O mere irratsionalnosti $\ln 3$”, Dokl. RAN, 417:6 (2007), 753–755 | MR | Zbl

[6] E. B. Tomashevskaya, O diofantovykh priblizheniyakh znachenii nekotorykh analiticheskikh funktsii, Dis. $\dots$ kand. fiz.-matem. nauk, BGTU, Bryansk, 2009

[7] C. Viola, W. Zudilin, “Hypergeometric transformations of linear forms in one logarithm”, Funct. Approx. Comment. Math., 39, Part 2 (2008), 211–222 | MR | Zbl

[8] R. Marcovecchio, “The Rhin–Viola method for $\log 2$”, Acta Arith., 139:2 (2009), 147–184 | DOI | MR | Zbl

[9] Yu. V. Nesterenko, “O pokazatele irratsionalnosti chisla $\ln 2$”, Matem. zametki, 88:4 (2010), 550–565

[10] M. Hata, “Rational approximations to $\pi$ and some other numbers”, Acta Arith., 63:4 (1993), 335–349 | MR | Zbl

[11] M. V. Fedoryuk, Metod perevala, Nauka, M., 1977 | MR | Zbl