The generating function )
for the associated Laguerre polynomials is a mathematical tool that, when expanded as a power series, produces associated Laguerre polynomials as its coefficients in terms of a variable.
It is defined as
=\frac{e^{-\frac{xt}{1-t}}}{(1-t)^{m+1}}=\sum_{n=0}^{\infty}L_n^m(x)t^n\;\;\;\;\;\;\;\;448)
To prove eq448, we differentiate the generating function for the Laguerre polynomials (see eq430) 
times with respect to 
to give
^mt^m}{(1-t)^{m+1}}e^{-\frac{xt}{1-t}}=\sum_{n=0}^{\infty}\frac{d^mL_n(x)}{dx^m}t^n)
When 
, the terms in the summation equal zero. So,
^mt^m}{(1-t)^{m+1}}e^{-\frac{xt}{1-t}}=\sum_{n=m}^{\infty}\frac{d^mL_n(x)}{dx^m}t^n)
From eq443, =(-1)^m\frac{d^mL_n(x)}{dx^m})
, which when substituted in the above equation yields,
^mt^m}{(1-t)^{m+1}}e^{-\frac{xt}{1-t}}=\sum_{n=m}^{\infty}(-1)^mL_{n-m}^m(x)t^n)
Letting 
gives
^mt^m}{(1-t)^{m+1}}e^{-\frac{xt}{1-t}}=t^m(-1)^m\sum_{k=0}^{\infty}L_{k}^m(x)t^k)
which rearranges to eq448.