Rational approximations for values of derivatives of the Gamma function

Abstract:

The arithmetic nature of Euler’s constant $\gamma$ 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 $a_n$ and $b_n$ such that $a_n/b_n$ converges sub-exponentially to $\gamma$, viewed as $-\Gamma ’(1)$, where $\Gamma$ is the usual Gamma function. Although this is not yet enough to prove that $\gamma \not \in \mathbb {Q}$, 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 $\mathbb {Q}(z)$ whose height grows like the factorial and that converges sub-exponentially to $\log (z)+\gamma$ for any complex number $z\in \mathbb {C}\setminus (-\infty ,0]$, where $\log$ is defined by its principal branch. We also show how our approach yields in theory rational approximations of numbers related to $\Gamma ^{(s)}(1)$ for any integer $s\ge 1$. In particular, we construct a sixth order recurrence which provides simultaneous rational approximations (of factorial height) converging sub-exponentially to the numbers $\gamma$ and $\Gamma ''(1)-2\Gamma ’(1)^2=\zeta (2)-\gamma ^2.$