# Nesting differential operators

Suppose I define differential operators $$d=x\frac{\partial}{\partial x}$$ and $$L=g(x)d^4$$ (so that $$L(f)=g(x)(x\frac{\partial}{\partial x})^4f$$ ) and I want to expand the differential operator $$L=a_0+a_1 \frac{\partial}{\partial x}+ a_2 \frac{\partial^2}{\partial x^2} + \cdots$$. I tried doing the following but it doesn't work.

d[f_] := x D[f,x]
g[x_] := 2 x
L[f_] := g[x] Nest[d,f,4]
Expand[L[g[x]]]

• Maybe a starting point: d[f_] := # Derivative[1][f][#] &. – b.gates.you.know.what Feb 4 at 13:28
• Fourth Derivative is D[f[x],{x,4}] – nilo de roock Feb 4 at 14:05
• It seems that everything is correct. L[g[x]] == L[2 x] == 4 x^2, because g[x] is defined to be 2x. Expand[L[h[x]] looks like what you expected. – aooiiii Feb 4 at 14:42
• Section "Some noncommutative algebraic manipulation" here might have usable approaches. – Daniel Lichtblau Feb 4 at 14:59