Any closed form for this expression?$ \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}$

Question

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!!!

Answer 1

Let's find the hypergeometric form, using the method from https://en.wikipedia.org/wiki/Generalized_hypergeometric_function:

$$\sum_{k=0}^{k=n}\frac{\lambda^{m-n+2k}\sqrt{n!\,m!}}{k!\,(m-n+k)!(n-k)!}= \sqrt{ \binom{m}{n} } \frac{\lambda^{m-n}}{\sqrt{(m-n)!}}\sum_{k=0}^n c_k \lambda^{2k}$$

Where:

$$c_0=1$$

$$\frac{c_{k+1}}{c_k}=\frac{\lambda^2 (n-k)}{(k+1)(m-n+k+1)}=\frac{k-n}{k+m-n+1}\frac{-\lambda^2}{k+1}$$

Which (by definition) makes the sum equal to:

$$\sum_{k=0}^{k=n}\frac{\lambda^{m-n+2k}\sqrt{n!\,m!}}{k!\,(m-n+k)!(n-k)!}= \sqrt{ \binom{m}{n} } \frac{\lambda^{m-n}}{\sqrt{(m-n)!}}{_1 F_1}(-n;m-n+1;-\lambda^2)$$

This is known as https://en.wikipedia.org/wiki/Confluent_hypergeometric_function and has many special properties listed on this page. There might be a better closed form for the integer parameters.

Answer 2

$$\frac{\Gamma (m+1) \lambda ^{m-n} \, _1\tilde{F}_1\left(-n;m-n+1;-\lambda ^2\right)}{\sqrt{\Gamma (m+1) \Gamma (n+1)}}$$