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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Rational approximations for values of derivatives of the Gamma function
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by Tanguy Rivoal PDF
Trans. Amer. Math. Soc. 361 (2009), 6115-6149 Request permission

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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Additional Information
  • Tanguy Rivoal
  • Affiliation: Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d’Hères cedex, France
  • MR Author ID: 668668
  • Received by editor(s): February 28, 2008
  • Published electronically: June 25, 2009
  • © Copyright 2009 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 6115-6149
  • MSC (2000): Primary 11J13; Secondary 33C45, 33F10, 39A11
  • DOI: https://doi.org/10.1090/S0002-9947-09-04905-8
  • MathSciNet review: 2529926