Application of Padé approximation to Euler's constant and Stirling's formula

Marc Prévost (1) , Tanguy Rivoal (2)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
2 IF - Institut Fourier

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

Domaines: Théorie des nombres [math.NT]; Analyse numérique [math.NA]

Marc Prévost, Tanguy Rivoal. Application of Padé approximation to Euler's constant and Stirling's formula. Ramanujan Journal, 2021, 54 (1), pp.177-195. ⟨10.1007/s11139-019-00201-9⟩. ⟨hal-02017852v3⟩