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?

Last edited:

fresh_42

Mentor
2018 Award
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.

"How to prove differential property of homogeneous function"

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