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Using the scattering theory on the hyperbolic space $\mathbb{H}^n$, we givethe explicit formulas of the fractional GJMS operators $P_{\gamma}$ for all$\gamma\in(0,\frac{n}{2})\setminus\mathbb{N}$ on $\mathbb{H}^n$.These$P_{\gamma}$ for $\gamma\in(0,\frac{n}{2})\setminus\mathbb{N}$ are neitherconformal to the fractional Laplacians on $\mathbb{R}^n_{+}$ nor on$\mathbb{B}^n$ in $\mathbb{R}^{n}$ though $P_{\gamma}$ are conformal to$(-\Delta)^{\gamma}$ via half space model and ball model of hyperbolic spaceswhen $\gamma\in\mathbb{N}$. To circumvent this, we introduce another family offractional operators $\tilde{P}_{\gamma}$ on $\mathbb{H}^n$ which are conformalto the fractional Laplacians on $\mathbb{R}^n_{+}$ and $\mathbb{B}^n$. It isworthwhile to note that $\tilde{P}_{\gamma}\not =P_{\gamma}$ unless $\gamma$ isan integer. We establish the fractional Poincar\'e-Sobolev inequalitiesassociated with both $P_{\gamma}$ and $\tilde{P}_{\gamma}$ on $\mathbb{H}^n$.In particular, when $n\geq 3$ and $\frac{n-1}{2}\leq \gamma<\frac{n}{2}$, weprove that the sharp constants in the $\gamma$-th order of Poincar\'e-Sobolevinequalities on the hyperbolic space associated with $P_{\gamma}$ and$\tilde{P}_{\gamma}$ coincide with the best $\gamma$-th order Sobolev constantin the $n$-dimensional Euclidean space $\mathbb{R}^n$. We also establishfractional Hardy-Sobolev-Maz'ya inequality on $\mathbb{R}^{n}_+$ and$\mathbb{B}^n$ and prove that the sharp constants in the $\gamma$-th orderHardy-Sobolev-Maz'ya inequalities on half space $\mathbb{R}^{n}_+$ and unitball $\mathbb{B}^n$ are the same as the best $\gamma$-th order Sobolevconstants in $\mathbb{R}^n$ when $n\geq 3$ and $\frac{n-1}{2}\leq\gamma<\frac{n}{2}$. Our methods crucially rely on the Helgason-Fourieranalysis on hyperbolic spaces and delicate analysis of special functions.

Explicit Formulas of Fractional GJMS operators on hyperbolic spaces and sharp fractional Poincar\'e-Sobolev and Hardy-Sobolev-Maz'ya inequalities