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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Rational approximations for values of derivatives of the Gamma function

Author: Tanguy Rivoal
Journal: Trans. Amer. Math. Soc. 361 (2009), 6115-6149
MSC (2000): Primary 11J13; Secondary 33C45, 33F10, 39A11
Published electronically: June 25, 2009
MathSciNet review: 2529926
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Abstract: The arithmetic nature of Euler's constant $ \gamma$ is still unknown and even getting good rational approximations to it is difficult. Recently, Aptekarev managed to find a third order linear recurrence with polynomial coefficients which admits two rational solutions $ a_n$ and $ b_n$ such that $ a_n/b_n$ converges sub-exponentially to $ \gamma$, viewed as $ -\Gamma'(1)$, where $ \Gamma$ is the usual Gamma function. Although this is not yet enough to prove that $ \gamma\not\in\mathbb{Q}$, it is a major step in this direction.

In this paper, we present a different, but related, approach based on simultaneous Padé approximants to Euler's functions, from which we construct and study a new third order recurrence that produces a sequence in $ \mathbb{Q}(z)$ whose height grows like the factorial and that converges sub-exponentially to $ \log(z)+\gamma$ for any complex number $ z\in\mathbb{C}\setminus (-\infty,0]$, where $ \log$ is defined by its principal branch. We also show how our approach yields in theory rational approximations of numbers related to $ \Gamma^{(s)}(1)$ for any integer $ s\ge 1$. In particular, we construct a sixth order recurrence which provides simultaneous rational approximations (of factorial height) converging sub-exponentially to the numbers $ \gamma$ and $ \Gamma''(1)-2\Gamma'(1)^2=\zeta(2)-\gamma^2.$

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Tanguy Rivoal
Affiliation: Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d’Hères cedex, France

Keywords: Euler's constant, rational approximations, Pad\'e approximants, linear recurrences, Birkhoff--Trjitzinsky theory
Received by editor(s): February 28, 2008
Published electronically: June 25, 2009
Article copyright: © Copyright 2009 American Mathematical Society