The arithmetic nature of Euler's constant is still unknown and even getting good rational approximations to it is difficult. Recently, Aptekarev managed to find a third order linear recurrence with polynomial coefficients which admits two rational solutions and such that converges sub-exponentially to , viewed as , where is the usual Gamma function. Although this is not yet enough to prove that , it is a major step in this direction.

In this paper, we present a different, but related, approach based on simultaneous Padé approximants to Euler's functions, from which we construct and study a new third order recurrence that produces a sequence in whose height grows like the factorial and that converges sub-exponentially to for any complex number , where is defined by its principal branch. We also show how our approach yields in theory rational approximations of numbers related to for any integer . In particular, we construct a sixth order recurrence which provides simultaneous rational approximations (of factorial height) converging sub-exponentially to the numbers and