Inner product of position eigenstates and coherent states

I was looking for the inner product between the position eigenstate $|xrangle$ and the coherent state $|zrangle$. I thought I’d go about it by expressing the position eigenstate in the number basis $|nrangle$. For that, I expressed the position operator $$hat{x}=dfrac{1}{sqrt{2}}(hat{a}+hat{a}^{dagger})$$. Using the fact that,
begin{align}
hat{a}|nrangle&=sqrt{n}|n-1rangle
hat{a}^{dagger}|nrangle&=sqrt{n+1}|n+1rangle
end{align} and assuming,
begin{align}
|xrangle&=sumlimits_{n=0}^{infty}c_n|nrangle
x|xrangle&=x|xrangle
end{align}
I finally obtained the recurrence relations:
begin{align}
c_1&=xc_0
c_{n+1}sqrt{n+1}+c_{n-1}sqrt{n}&=xc_n,n>0,ninmathbb{N}
end{align}
Can anyone help me in solving these recurrence relations?