On a theorem in the differential calculus. (Q1527451)

Eine allgemeine Formel für \[ f_n\left(\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\cdots,\frac{\partial}{\partial x_p}\right)\,F\{\varphi(x_1,x_2,x_3,\dots,x_p)\}, \] wo \(f_n\) eine rationale ganze homogene Function \(n^{\text{ten}}\) Grades in den Differentialoperatoren ist. Der wichtigste besondere Fall ist \[ \begin{multlined} f_n\left(\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\cdots,\frac{\partial}{\partial x_p}\right)F(x_1^2+x_2^2+\cdots+x_p^2) =\\ \{2^n\frac{d^nF}{d(\varrho^2)^n} + \frac{2^{n-2}}{1!}\frac{d^{n-1}F}{d(\varrho^2)^{n-1}}\nabla^2 + \frac{2^{n-4}}{2!}\frac{d^{n-2}F}{d(\varrho^2)^{n-2}}\nabla^4 +\cdots\} \times f_n(x_1,x_2,\dots,x_p),\end{multlined} \] wo \[ \nabla^2 = \frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial x_2^2} +\cdots+ \frac{\partial^2}{\partial x_p^2},\;\varrho^2 = x_1^2+x_2^2+\cdots+x_p^2 \]