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 Pade 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 Apery’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.