TITLE: Curious gamma identity
QUESTION [5 upvotes]: I found the following curious identity for the gamma function on Wikipedia for which I'd like to know some references (proof, history, etc).
The identity is as follows: $$\Gamma(t) = x^t \sum_{n=0}^\infty \frac{L_n^{(t)}(x)}{t+n}$$
Here $L_n^{(t)}(x)$ are the generalized Laguerre polynomials, and the expression seems to be valid for $\mathrm{Re}(t) < \frac12$. (That's all which is specified on Wiki). 
EDIT: There are now two very helpful answer, but before accepting one of them I'd like to broaden the question a little bit further in asking for a representation theoretic interpretation of the gamma function identity.
Thanks a lot in advance!

REPLY [2 votes]: A reference:  Chaudhry, M.A. et al. Asymptotics and closed form of a generalized incomplete gamma function.