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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]

[tex]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)[/tex]

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

[tex]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)[/tex]

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

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