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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
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 \({\widehat{F}}(z):=\sum _{k=0}^\infty m_k z^k \in {\mathbb {K}}[[z]]\).
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].
References
Alladi, K., Robinson, M.L.: Legendre polynomials and irrationality. Crelle’s J. 318, 137–155 (1980)
Apéry, R.: Irrationality of \(\zeta (2)\) and \(\zeta (3)\). Astérisque 61, 11–13 (1979)
Aptekarev, A.I.: Rational Approximants for Euler Constant and Recurrence Relations, Sovremennye Problemy Matematiki (“Current Problems in Mathematics”), vol. 9. MIAN (Steklov Institute), Moscow (2007)
Askey, R., Wilson, J.: A set of hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 13(4), 651–655 (1982)
Brezinski, C.: Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, vol. 50. Birkhauser, Basel (1980)
Campbell, R.: Les Intégrales eulériennes et leurs applications: étude approfondie de la fonction gamma. Dunod, Paris (1966)
Miller, P.D.: Applied Asymptotic Analysis. Graduate Studies in Mathematics, vol. 75. American Mathematical Society, Providence, RI (2006)
Prévost, M.: A new proof of the irrationality of \(\zeta (2)\) and \(\zeta (3)\) using Padé approximants. J. Comput. Appl. Math. 67(2), 219–235 (1996)
Prévost, M.: Legendre modified moments for Euler’s constant. J. Comput. Appl. Math. 219(2), 484–492 (2008)
Prévost, M.: Remainder Padé approximants for the Hurwitz zeta function. Results Math. 74, 22 (2019)
Rivoal, T.: Nombres d’Euler, approximants de Padé et constante de Catalan. Ramanujan J. 11(2), 199–214 (2006)
Rivoal, T.: Rational approximations for values of derivatives of the Gamma function. Trans. Am. Math. Soc. 361, 6115–6149 (2009)
Touchard, J.: Nombres exponentiels et nombres de Bernoulli. Can. J. Math. 8, 305–320 (1956)
Wilson, J.: Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 11, 690–701 (1980)
Acknowledgements
We thank the referee for her or his careful reading of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Prévost, M., Rivoal, T. Application of Padé Approximation to Euler’s constant and Stirling’s formula. Ramanujan J 54, 177–195 (2021). https://doi.org/10.1007/s11139-019-00201-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-019-00201-9
Keywords
- Euler’s constant
- Gamma function
- Digamma function
- Stirling’s formula
- Bernoulli numbers
- Padé approximants
- Orthogonal polynomials