Definition: A Cauchy-Euler operator is a differential operator of the form\[
p(x)\frac{d}{dx},
\]
where $p(x)$ is a polynomial.
Proposition: Let\[\left(x\frac{d}{dx}\right)^m=a_{m,1}x\frac{d}{dx}+a_{m,2}x^2\frac{d^2}{dx^2}+\cdots+a_{m,m-1}x^{m-1}\frac{d^{m-1}}{dx^{m-1}}+a_{m,m}x^m\frac{d^m}{dx^m},~m=1,\ldots,n\]
Then we have the recurrence relation:
\begin{cases}
a_{m,1}=1,~a_{m,m}=1, & m=1,\ldots,n\\
a_{m,k}=ka_{m-1,k}+a_{m-1,k-1}, & k=2,\ldots,m-1
\end{cases}
