  This document is a companion for the Maple program : Discrete series and
K-types for U(p,q) available on:http://www.math.jussieu.fr/~vergne We explain
an algorithm to compute the multiplicities of an irreducible representation of
U(p)x U(q) in a discrete series of U(p,q). It is based on Blattner's formula.
We recall the general mathematical background to compute Kostant partition
functions via multidimensional residues, and we outline our algorithm. We also
point out some properties of the piecewise polynomial functions describing
multiplicities based on Paradan's results.
