It's a standard result that the three well-known quasiprobability distributions can all be expressed in terms of the "$s$-ordered characteristic functions" as $$ W(\alpha) = \int\frac{d^2\eta}{\pi^2} \chi_W(\eta) e^{\alpha\bar\eta-\bar\alpha\eta}, \\ P(\alpha) = \int\frac{d^2\eta}{\pi^2} \chi_N(\eta) e^{\alpha\bar\eta-\bar\alpha\eta}, \\ Q(\alpha) = \int\frac{d^2\eta}{\pi^2} \chi_A(\eta) e^{\alpha\bar\eta-\bar\alpha\eta}. $$ where $\chi_W,\chi_A,\chi_N$ are the symmetrically, antinormally, and normally ordered characteristic functions, respectively. These can also be expressed concisely as $\chi_W=\chi_0$, $\chi_N=\chi_1$, $\chi_A=\chi_{-1}$, and (with some abuse of notation) $$\chi_s(\eta) = \operatorname{tr}[\rho\exp(\eta a^\dagger-\bar\eta a + s|\eta|^2/2)].$$ The above formulation just begs the question: what happens for $s\notin\{-1,0,1\}$? If we were to consider the functions $$W_s(\alpha) \equiv \int\frac{d^2\eta}{\pi^2} \chi_s(\eta) e^{\alpha\bar\eta-\bar\alpha\eta} =\int\frac{d^2\eta}{\pi^2} \operatorname{tr}\left[\rho\exp\left((\alpha-a)\bar\eta-(\bar\alpha-a^\dagger)\eta+\frac s2 |\eta|^2\right)\right],$$ for $s\neq -1,0,1$, would these still be well-defined quasiprobability distributions? Would they correspond to some other "weirder" kind of operator ordering?
This question also stems from finding some interesting expressions in (Kahill and Glauber 1969, PhysRev.177.1857). They show that, defining (Eq. 6.6) $$T(\alpha,s) = \int\frac{d^2\xi}{\pi} \exp\left[ (\alpha-a)\bar\xi- (\bar\alpha-a^\dagger)\xi+\frac{s}{2}|\xi|^2\right],$$ one can derive the explicit expression (Eq. 6.24): $$T(\alpha,s) = \frac{2}{1-s} \left(\frac{s+1}{s-1}\right)^{(a^\dagger-\bar\alpha)(a-\alpha)}.$$ Note that the quasiprobability distribution I defined before are connected to these via $W_s(\alpha)=\frac1\pi\operatorname{tr}[\rho T(\alpha,s)] $ (and the $\pi$ factor is just due to different conventions in the employed Fourier transforms). Now, of course, this expression remains rather unwieldy to work with, and its singularities are to be handled with appropriate limits, but it still seems natural to wonder what happens if you plug different values of $s$ in it.
