Laguerre's gf

[Karol]

    Does anyone know closed form  of the following generating function
     for the ordinary Laguerre polynomials:

    G(M;T,x)= sum( t^n*LaguerreL(n,x)/((n!)^M), n=0..infinity), for 
M=2,3...          ?
 
     Thanks in advance,    Karol A. Penson