Husmi Q-functions attached to hyperbolic Landau levels

We are concerned with a phase-space probability distribution which is known as Husimi $Q$-function of a density operator with respect to a set of coherent states $\vert\widetilde{\kappa}_{z,B,R,m}\rangle$ attached to an $m$th hyperbolic Landau level and labeled by points $z$ of an open disk of radius $R$, where $B>0$ is proportional to a magnetic field strength. For a density operator representing a projector on a Fock state $\left\vert j\right\rangle$ we obtain the $Q_{j}$ distribution and discuss some of its basic properties such as its characteristic function and its main statistical parameters. We achieve the same program for the thermal density operator (mixed states) of the isotonic oscillator for which we establish a lower bound for the associated thermodynamical potential. We recover most of the results of the Euclidean setting (flat case) as the parameter $R$ goes to infinity by making appeal to asymptotic formulas involving orthogonal polynomials and special functions. As a tool, we establish a summation formula for the special Kamp\'{e} de F\'{e}riet function $\digamma _{2:0:0}^{1:2:2}$.