  The inner-outer part factorisation of analytic representations in the unit
disk is used for an effective characterisation of the number-phase statistical
properties of a quantum harmonic oscillator. It is shown that the factorisation
is intimately connected to the number-phase Weyl semigroup and its properties.
In the Barut-Girardello analytic representation the factorisation is
implemented as a convolution. Several examples are given which demonstrate the
physical significance of the factorisation and its role for quantum statistics.
In particular, we study the effect of phase-space interference on the
factorisation properties of a superposition state.
