An effective way of computing in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). Or we can use forward recurrence, with an initial value obtained e.g. from (5.7.3). For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3).
Similarly for , , and the polygamma functions.
For the computation of the -gamma and -beta functions see Gabutti and Allasia (2008).