On a continued fraction expansion for Euler's constant

Details On a continued fraction expansion for Euler's constant On a continued fraction expansion for Euler's constant

2010-10-07

arxiv.org/abs/1010.1420v1

Mathematics

Number Theory

21 pages

Scientific paper

Recently, A.~I.~Aptekarev and his collaborators found a sequence of rational approximations to Euler's constant $\gamma$ defined by a third-order homogeneous linear recurrence. In this paper, we give a new interpretation of Aptekarev's approximations to $\gamma$ 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 $\gamma$ with rational coefficients. Using this construction we find new rational approximations to $\gamma$ generated by a second-order inhomogeneous linear recurrence with polynomial coefficients. This leads to a continued fraction (though not a simple continued fraction) for Euler's constant. It seems to be the first non-trivial continued fraction expansion convergent to Euler's 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.

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