# Applying the position operator several times to a harmonic oscillator state $\hat x^m |n\rangle =$ ______? [duplicate]

The position operator can be expressed in terms of the harmonic oscillator ladder operators

$$\hat x = \hat a + \hat a^\dagger,$$

in natural units. Therefore we have

$$\hat x |n\rangle = \frac{\hat a + \hat a^\dagger}{\sqrt 2} |n\rangle = \sqrt{\frac{n}{2}}|n-1\rangle + \sqrt{\frac{n+1}{2}}|n+1\rangle.$$

My question is if there is a way to write up the state resulting from applying the position operator an arbitrary number of times? Due to the above relation we can write

$$\hat x^m |n_0\rangle = \sum_{n=n_0-m}^{n_0+m} c_n |n\rangle,$$

where only even or odd coefficients will be non-zero, and where we have assumed $$n_0>m$$. But is there a way we can write a general expression for $$c_n$$ as a function of $$m,n_0$$ by using the recurrence relation above?

A foolproof approach is doing this in position representation (where it is just multiplication by $$x^m$$) using your favorite methods of dealing with Hermit polynomials (e.g., Schiff has a clear presentation of using the generating function, and thus reducing everything to Gaussian integrals.)
Another way is by evaluating first $$e^{\lambda(a^\dagger + a)}|n\rangle\propto e^{\lambda a^\dagger}e^{\lambda a}|n\rangle,$$ where the proportionality coefficient is obtained with Baker-Hausdorff formula. One can then expand result in powers of $$\lambda$$. In my memory, one ends up with Laguerre polynomials or something like that - so don't expect it to be simple, whatever method one uses.
Finally, one could try induction by evaluating $$x|n\rangle$$, $$x^2|n\rangle$$, etc. and guessing the general rule.