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

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 Pade 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 Apery'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.