  We introduce three representation formulas for the fractional $p$-Laplace
operator in the whole range of parameters $0<s<1$ and $1<p<\infty$. Note that
for $p\ne 2$ this a nonlinear operator. The first representation is based on a
splitting procedure that combines a renormalized nonlinearity with the linear
heat semigroup. The second adapts the nonlinearity to the Caffarelli-Silvestre
linear extension technique. The third one is the corresponding nonlinear
version of the Balakrishnan formula. We also discuss the correct choice of the
constant of the fractional $p$-Laplace operator in order to have continuous
dependence as $p\to 2$ and $s \to 0^+, 1^-$.
  A number of consequences and proposals are derived. Thus, we propose a
natural spectral-type operator in domains, different from the standard
restriction of the fractional $p$-Laplace operator acting on the whole space.
We also propose numerical schemes, a new definition of the fractional
$p$-Laplacian on manifolds, as well as alternative characterizations of the
$W^{s,p}(\mathbb{R}^n)$ seminorms.
