I am looking for a closed form of this expression. If you have seen something like this or remember something similar, please let me know. My sincere thank!
$$ \sum_{k=0,\,l=0}^{k=n,\,l=m}\frac{\lambda^{l+k}}{k!\,l!}\sqrt{\frac{n!\,m!}{(n-k)!(m-l)!}}\delta_{n-k,\,m-l}$$
or for the case where $m>n$,
$$ \sum_{k=0}^{k=n}\frac{\lambda^{m-n+2k}\sqrt{n!\,m!}}{k!\,(m-n+k)!(n-k)!}$$
I would like to have a closed form of this expression in terms of $\lambda,m,n$.
For more information, this expression comes from $\langle m|e^{i\hat{x}}|n\rangle$ in quantum mechanics where $|n\rangle$ is the hamiltonian eigenstate of a simple harmonic oscillator and $\hat x$ is the position operator.
Thank you very much!!!