Rational approximations to values of the digamma function and a conjecture on denominators

We explicitly construct rational approximations to the numbers ln(b) - psi(a + 1), where psi is 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 +/- gamma. The proposed construction is compared 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).