Rational approximations for values of derivatives of the Gamma function

Abstract

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 <!-- MATH $-\Gamma'(1)$ --> , where is the usual Gamma function. Although this is not yet enough to prove that <!-- MATH $\gamma\not\in\mathbb{Q}$ --> , it is a major step in this direction.