Sum over (squares of) Laguerre Polynomials

Question

I'm looking for a closed form of the sum \begin{equation} \sum_{n=0}^\infty \frac{n!}{(n+k)!} (L_n^k(x))^2 t^n, \end{equation} where $L_n^k(x)$ are the Laguerre Polynomials. I have been looking for some time and only found equations that are pretty close, but not exactly what I need (like here).

Answer 1

Go the wikipedia article for Laguerre polynomials and search for 'Hardy-Hille' formula near the end of it. Put $\alpha=k$ and $y=x.$