I have a sequence of differential operators given by
$H_n = x \frac{d}{dx} + n$
where $n=1,2,...$ and I would like to construct the operator
$\hat{O}(n) = \frac{1}{n!}H_nH_{n-1}...H_2H_1$
as a function of the integer $n$. I know how to compose operators, and in particular how to compose the same operator $n$ times using, say, the Nest function, e.g. for $nH_1$ n-times I'd write
h[x_, n_] := x D[#, x] + n # &
ohat1[n_] := 1/n! Nest[h[x, 1], #, n] &
which would give me $\hat{O}_1(n) = (H_1)^n/n!$, but I haven't been able to generalise this to the case I'm interested in.
Aside: if this is even possible would mathematica then be able to deal with the case $n=\infty$ when acting on a specific function?
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button (or indentation of every line of code with four spaces). $\endgroup$ – Feyre Sep 1 '16 at 8:28Product
don't you? Try1/n! Product[h[x, i], {i, 1, n}]
$\endgroup$ – Quantum_Oli Sep 1 '16 at 8:56