Given two positive integers \(a\) and \(b\) with \(0<a<b<1\), the author produces an explicit sequence \((p_n/q_n)_{n\geq 0}\) of rational numbers converging towards \(\Gamma(a/b)^b\). This sequence arises from an integral \(I_n\) which is similar to integrals occurring in the rational interpolation of meromorphic functions, like Hurwitz zeta function. The author proves that his integrals are of the form \(I_n=q_n\Gamma(a/b)^b-p_n\) where \(p_n\) and \(q_n\) are rational numbers. He gives explicit estimates for the denominator of \(p_n\) and \(q_n\), and for the size of \(|p_n|\), \(|q_n|\) and \(|I_n|\). He also shows that the three sequences \((p_n)_{n\geq 0}\), \((q_n)_{n\geq 0}\) and \((I_n)_{n\geq 0}\) satisfy the same linear recurrence sequence with coefficients in \(\mathbb Z[n]\), which he explicitly gives for \(a=1\) and \(b=2\), \(3\) and \(4\). He compares his results with sequences of rational \(G\)-approximations of a real number, related with Siegel’s \(G\)-functions. He raises new open questions on the connection between numbers admitting rational \(G\)–approximations on the one hand and periods on the other hand.