Rational Approximations to Values of the Digamma Function and a Conjecture on Denominators

We obtain explicit constructions for rational approximations to the numbers $\ln(b)-\psi(a+1)$, where $\psi$ defines the logarithmic derivative of the Euler gamma function. We prove formulas expressing the numerators and the denominators of the approximations in terms of hypergeometric sums. This generalizes the Aptekarev construction of rational approximations for the Euler constant $\gamma$. As a consequence, we obtain rational approximations for the numbers $\pi/2\pm\gamma$. The proposed construction is compared with with rational Rivoal approximations for the numbers $\gamma+\ln(b)$. We verify assumptions put forward by Rivoal on the denominators of rational approximations to the numbers $\gamma+\ln(b)$ and on the general denominators of simultaneous approximations to the numbers $\gamma$ and $\zeta(2)-\gamma^2$.