 Generating series of the Kostant partition function

The Kostant partition function counts nonnegative integer combinations of the positive roots of a root system, without ordering the summands. Multiply the formal geometric series ${\sum }_{k\ge 0}{e}^{-k\alpha }$ for every positive root. The coefficient of $e^{-\nu }$ counts precisely these combinations. Every coefficient is finite because a linear functional strictly positive on the positive roots bounds every participating integer coefficient. The sign of each exponent in the cancelling product must agree with that of this series.