The fractional $p$-Laplacian on hyperbolic spaces

We present three equivalent definitions of the fractional $p$-Laplacian $(-{\mathrm{\Delta }}_{{\mathbb{H}}^n}{)}_p^s$, $0<s<1$, $p>1$, with normalizing constants, on hyperbolic spaces. The explicit values of the constants enable us to study the convergence of the fractional $p$-Laplacian to the $p$-Laplacian as $s\to 1^{-}$.