Rational approximations for values of the digamma function and a denominators conjecture

In 2007, A.I.Aptekarev and his collaborators discovered a sequence of rational approximations to Euler’s constant $\gamma$ defined by a linear recurrence. In this paper, we generalize this result and present an explicit construction of rational approximations for the numbers $\ln (b)-\psi (a+1),$ $a, b\in {\mathbb Q},$ $b>0,a>-1,$ where $\psi$ defines the logarithmic derivative of the Euler gamma function. We prove exact formulas for denominators and numerators of the approximations in terms of hypergeometric sums. As a consequence, we get rational approximations for the numbers $\pi \mathrm{/}2\pm \gamma \mathrm{.}$ We compare the results obtained with those of T. Rivoal for the numbers $\gamma +\ln (b)$ and prove denominators conjectures proposed by Rivoal for denominators of rational approximations for $\gamma +\ln (b)$ and common denominators of simultaneous approximations for the numbers $\gamma$ and $\zeta (2)-{\gamma }^2\mathrm{.}$