Find the eigenstates of a basis in terms of those of another basis?

1. The problem statement, all variables and given/known data

This isn’t exactly a homework question, but I figured this would be the best subforum for this sort of thing. For the sake of a concrete example, let’s just say my question is:

Express the position operator’s eigenstates in terms of the number operator’s eigenstates.

2. Relevant equations

The number operator is given by
[tex]\hat{N} = \hat{a}^\dagger\hat{a}.[/tex]
It has eigenvalues and eigenstates
[tex]\hat{N}|n\rangle = n|n\rangle.[/tex]
The position operator is given by
[tex]\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}\Big(\hat{a}^\dagger+\hat{a}\Big) \equiv \gamma\Big(\hat{a}^\dagger+\hat{a}\Big).[/tex]
The action of the position operator on one of the number eigenstates is
[tex]\hat{x}|n\rangle = \gamma\Big(\hat{a}^\dagger|n\rangle + \hat{a}|n\rangle\Big)
=\gamma\Big(\sqrt{n+1}|n+1\rangle + \sqrt{n}|n-1\rangle\Big).[/tex]

3. The attempt at a solution

We’d like to find eigenstates of [itex]\hat x[/itex], that is, states [itex]|x\rangle[/itex] satisfying
[tex]\hat x|x\rangle = x|x\rangle.[/tex]
The number basis is complete, so whatever these states we’re looking for might be, they are representable as a linear combination of the number states:
[tex]|x\rangle = \sum_n |n\rangle\langle n|x\rangle;[/tex]
I just need to know the coefficients [itex]C_n \equiv \langle n|x\rangle[/itex].

If I look again at the action of [itex]\hat x[/itex],
[tex]\hat x|x\rangle = \sum_n \hat x|n\rangle\langle n|x\rangle \\ \qquad = \sum_n \gamma\Big(\sqrt{n+1}|n+1\rangle + \sqrt{n}|n-1\rangle\Big) \langle n|x\rangle.[/tex]
Suppose I attack this equation from the left with a particular [itex]\langle n’|[/itex]. Then I get
[tex]x\langle n’|x\rangle = \sum_n \gamma\Big(\sqrt{n+1}\langle n’|n+1\rangle + \sqrt{n}\langle n’|n-1\rangle\Big) \langle n|x\rangle \\
\qquad = \gamma\Big(\sqrt{n’}\langle n’-1|x\rangle + \sqrt{n’+1}\langle n’+1|x\rangle\Big),[/tex]
which allows me to develop a recurrence relationship
[tex]C_{n+1} = \frac{x}{\gamma\sqrt{n+1}}C_n-\frac{\sqrt{n}}{\sqrt{n+1}}C_{n-1}.[/tex]

But I don’t understand what this says. What is that [itex]x[/itex] doing in there? What does that even mean? And can I use this recurrence relation to get a closed-form answer? (At the very least I’d need to calculate [itex]C_0[/itex] and [itex]C_1[/itex] explicitly, which I don’t know how to do.)

Also, I think at some point the position-space wavefunctions ought to come into it:
[tex]\langle x’|n\rangle = \sqrt[4]{\frac{m\omega}{\pi\hbar}}\frac{1}{\sqrt{2^nn!}}H_n\Big(\sqrt{\frac{1}{ \sqrt{2}\gamma} x}\Big)e^{-x^2/2\gamma^2}[/tex]
[tex]\langle x’|x\rangle = \delta(x-x’).[/tex]

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