I have discovered two interesting summation identities for Laguerre polynomials (although my derivation is obscure) and can't find them mentioned anywhere in my usual online resources. My background is physics, so I don't know better than that. As it's most improbable that something as elegant would not be known, could anyone please point me to any source where these, or some closely related ones, could be found, in case I need to cite this in my future work?
1) $$\sum_{k=0}^n L_n^{(k)}(0) L_k(x) = \frac{x^n}{n!}$$
2) $$\sum_{k=0}^n L_n^{(k)}(2\xi)_{\xi=0} L_k(2x) = (-1)^n L_n(4x)$$
In both cases, the $^{(k)}$ represents a $k$-th derivative.
Equivalent form without derivatives:
1) $$\sum_{k=0}^n (-1)^k L_{n-k}^k(0) L_k(x) = \frac{x^n}{n!}$$
2) $$\sum_{k=0}^n (-2)^k L_{n-k}^k(0) L_k(2x) = (-1)^n L_n(4x)$$