I am trying to prove the following equality: \begin{equation*} \sum_{k=0}^{\min(m,n)}\frac{(-1)^k}{(m-k)!(n-k)!k!}x^{\min(m,n)-k}=\frac{(-1)^{\max(m,n)}}{[\max(m,n)!]^2}L_{\max(m,n)}^{(|m-n|)}(x) \end{equation*} where $L_n^{(\alpha)}(x)$ denotes an associated Lauguerre polynomials. I am not even confident of its correctness, as tests with Mathematica are inconsistent with the result. In any case, what is behind the idea of the author? In particular, I can't understand how to pass from a sum till $\min(m,n)$ to a sum to an upper or equal value $\max(m,n)$, as this is the boundary of the corresponding seriex expansion of the associated Laguerre polynomials \begin{equation*} L_n^{(\alpha)}(x)=\sum_{i=0}^n \binom{n+\alpha}{n-i}\frac {x^i}{i!} \end{equation*} The expression appears in eqn. (5.16) at page 457 of this Physics paper (please note that the author himself slips in the denomination of the Laguerre polynomials, referring to them as "Legendre").
The only relation that came to my mind that could help the issue is \begin{equation*} (-x)^i\frac 1 {i!}L_j^{(i-j)}(x)=(-x)^j\frac 1 {j!}L_i^{(j-i)}(x) \end{equation*} in combination with $\min(m,n)= \max(m,n)-|m-n|$