Suppose $f:[0,\infty)\rightarrow\mathbb{R}$ satisfies: $$f(x)= \sum_{n=0}^\infty \hat{f}_n L_n(x),$$ for some $\hat{f}_0,\hat{f}_1,\dots\in\mathbb{R}$, where $L_n$ is the $n$th Laguerre polynomial for all $n\in\mathbb{N}$, and where the series converges absolutely for any $x\in[0,\infty)$ (pointwise).
Suppose further that for some $C,k,z\in[0,\infty)$, for all $x\in[z,\infty)$, $|f(x)|\le C x^k$.
What does this imply about the rate at which $\hat{f}_n\rightarrow 0$ as $n\rightarrow \infty$?
Wikipedia gives an asymptotic expansion for $L_n(x)$ for fixed $x$ which implies $L_n(x)=O(n^{-\frac{1}{4}})$ as $n\rightarrow\infty$. This means (I think) that it must at least be the case that $\hat{f}_n$ decays faster than $n^{-\frac{3}{4}}$, else the sums would not converge absolutely. Does my polynomial boundedness assumption imply a tighter bound?
(Aside: It is slightly surprising to me that $\hat{f}_n$ must decay faster than $n^{-\frac{3}{4}}$ for the series to converge pointwise, as $\mathcal{L}^2$ convergence just needs that $\hat{f}_n$ decays faster than $n^{-\frac{1}{2}}$. I know that Lp convergence does not imply pointwise convergence almost everywhere, but this seems insufficiently "weird" to be a source of counter-examples. Maybe my intuition is off.)
