Definition: A Cauchy-Euler operator is a differential operator of the formp(x)ddx,
where p(x) is a polynomial.
Proposition: Let(xddx)m=am,1xddx+am,2x2d2dx2+⋯+am,m−1xm−1dm−1dxm−1+am,mxmdmdxm, m=1,…,n
Then we have the recurrence relation:
{am,1=1, am,m=1,m=1,…,nam,k=kam−1,k+am−1,k−1,k=2,…,m−1