How to prove differential property of homogeneous function

[Prpan]

I came across of an equality which I have difficulty to understand. If f_n is a rational algebraic homogeneous function of degree n in the differential operators and if g_n is a regular non-differential homogeneous function of the same degree n, following equality takes place [Hobson: The theory of spherical and ellipsoidal harmonics, 1931]

$$f_n\left(\frac{\partial}{\partial{x_1}},\frac{\partial}{\partial{x_2}},...,\frac{\partial}{\partial{x_p}}\right)g_n(x_1,x_2,...,x_p)=g_n\left(\frac{\partial}{\partial{x_1}},\frac{\partial}{\partial{x_2}},...,\frac{\partial}{\partial{x_p}}\right)f_n(x_1,x_2,...,x_p)$$

Could someone help me to figure out how to obtain this equality?

 

[fresh_42]

We may consider only monomials, because differentiation is linear. The only terms which do not vanish are of the form $\dfrac{\partial^{n_{i_1}}}{\partial x_{i_1}}\ldots \dfrac{\partial^{n_{i_k}}}{\partial x_{i_k}} \left( x_{i_1}^{n_{i_1}}\ldots x_{i_k}^{n_{i_k}} \right)$ since all others either have a power less than $n_j$ or a variable which doesn't occur at all. The product rule doesn't matter, since we have partial derivatives where all other variables are treated as constants.