I may recommend here

https://math.stackexchange.com/questions/1797429/infinite-sum-involving-laguerre-polynomials

I recommend you to read :

Dan Dai, Mourad E.H.Ismail, Jun Wang. Asymptotics for Laguerre polynomials with large order and parameters. Journal of Approximation Theory 193 (2015) 4–19

In fact, in quantum mechanics, we need the associated Laguerre polynomials L_{n}^{m}(x), where n in {0,1,2,3,...,\infty} and m>-1 according to the definition of the associated Laguerre polynomials.

L_{n}^{m}(x) are defined by

L_{n}^{m}(x)=(-1)^{k} D^{k}_{x} L_{n+k}(x) , that mean k goes not to infinity.

See also the attached file