On a continued fraction expansion for Eulerʼs constant

Recently, A.I. Aptekarev and his collaborators found a sequence of rational approximations to Eulerʼs constant γ defined by a third-order homogeneous linear recurrence. In this paper, we give a new interpretation of Aptekarevʼs approximations in terms of Meijer G-functions and hypergeometric-type series. This approach allows us to describe a very general construction giving linear forms in 1 and γ with rational coefficients. Using this construction we find new rational approximations to γ generated by a second-order inhomogeneous linear recurrence with polynomial coefficients. This leads to a continued fraction (though not a simple continued fraction) for Eulerʼs constant. It seems to be the first non-trivial continued fraction expansion convergent to Eulerʼs constant sub-exponentially, the elements of which can be expressed as a general pattern. It is interesting to note that the same homogeneous recurrence generates a continued fraction for the Euler–Gompertz constant found by Stieltjes in 1895.