Generalization of Euler theorem for homogeneous polynomials

Question

Euler's theorem for homogeneous polynomials is well known. If $F:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is a homogeneous polynomial, then we have: $x_{1}\frac{\partial F }{\partial x_{1}} + ... + x_{n}\frac{\partial F }{\partial x_{n}} = deg(F)F$ . My question is: are there generalizations of this formula for higher order derivatives?