Husimi Q-Functions Attached to Hyperbolic Landau Levels

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