On some properties of number-phase Wigner function

It is shown that the number-phase Wigner function defines uniquely the respective density operator. Relations between the Glauber-Sudarshan distribution $\mathcal{P}(\alpha)$ and the number-phase Wigner function is found. This result is then generalised to the case of the Cahil-Glauber distributions $\mathcal{W}^{(s)}(\alpha)$, $-1\leq s \leq 1$.