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Application of Padé Approximation to Euler’s constant and Stirling’s formula

Abstract

The Digamma function \(\varGamma '/\varGamma \) admits a well-known (divergent) asymptotic expansion involving the Bernoulli numbers. Using Touchard-type orthogonal polynomials, we determine an effective bound for the error made when this asymptotic expansion is replaced by its nearly diagonal Padé approximant. By specialization, we obtain new fast converging sequences of approximations to Euler’s constant \(\gamma \). Even though these approximations are not strong enough to prove the putative irrationality of \(\gamma \), we explain why they can be viewed, in some sense, as analogs of Apéry’s celebrated sequences of approximations to \(\zeta (2)\) and \(\zeta (3)\). Similar ideas applied to the asymptotic expansion \(\log \varGamma \) enable us to obtain a refined version of Stirling’s formula.

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Notes

  1. 1.

    We make here a slight abuse of notation, i.e., \([k-1/k]_F(z)\) should be noted \([k-1/k]_{{\widehat{F}}}(z)\), where \({\widehat{F}}(z):=\sum _{k=0}^\infty m_k z^k \in {\mathbb {K}}[[z]]\).

  2. 2.

    The variable \(-z^2\) instead of z explains why the Padé approximants in this paper are evaluated at \(-\frac{1}{n^2}\) and not \(\frac{1}{n}\) as in [8].

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Acknowledgements

We thank the referee for her or his careful reading of the paper.

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Correspondence to T. Rivoal.

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Prévost, M., Rivoal, T. Application of Padé Approximation to Euler’s constant and Stirling’s formula. Ramanujan J (2020). https://doi.org/10.1007/s11139-019-00201-9

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Keywords

  • Euler’s constant
  • Gamma function
  • Digamma function
  • Stirling’s formula
  • Bernoulli numbers
  • Padé approximants
  • Orthogonal polynomials

Mathematics Subject Classification

  • Primary 11Y60
  • 41A21
  • Secondary 11J72
  • 41A60