Generating function for the Laguerre polynomials

The generating function $G(x,t)$ for the Laguerre polynomials is a mathematical tool that, when expanded as a power series, produces Laguerre polynomials as its coefficients in terms of a variable.

$G(x,t)=\frac{e^{-\frac{xt}{1-t}}}{1-t}=\sum_{n=0}^{\infty}L_n(x)t^n\;\;\;\;\;\;\;\;430$

Eq430 means that the cofficient of $t^n$ in the expansion of $\frac{e^{-\frac{xt}{1-t}}}{1-t}$ is $L_n(x)$ . To prove this, we expand the exponential term as a Taylor series:

$\frac{e^{-\frac{xt}{1-t}}}{1-t}=\frac{1}{1-t}\sum_{n=0}^{\infty}\frac{$-\frac{xt}{1-t}$^n}{n!}=\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}x^nt^n(1-t)^{-(n+1)}$

Expanding $(1-t)^{-(n+1)}$ as a binomial series gives

$\frac{e^{-\frac{xt}{1-t}}}{1-t}=\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}x^nt^n\sum_{k=0}^{\infty}\begin{pmatrix}-n-1\\k\end{pmatrix}(-t)^{k}$

Since $\begin{pmatrix}-n-1\\k\end{pmatrix}=\frac{(-n-1)!}{k!(-n-k-1)!}=\frac{(-n-1)(-n-2)\cdots (-n-k)}{k!}=\frac{(-1)^k(n+1)(n+2)\cdots(n+k)}{k!}=\frac{(-1)^k(n+k)!}{n!k!}$ ,

$\frac{e^{-\frac{xt}{1-t}}}{1-t}=\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}x^nt^n\sum_{k=0}^{\infty}\frac{(n+k)!}{n!k!}t^{k}=\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}(-1)^n\frac{(n+k)!}{k!(n!)^2}x^nt^{n+k}$

Letting $n+k=m$

$\frac{e^{-\frac{xt}{1-t}}}{1-t}=\sum_{n=0}^{\infty}\sum_{m=n}^{\infty}(-1)^n\frac{m!}{(m-n)!(n!)^2}x^nt^{m}\;\;\;\;\;\;\;\;431$

We now have a sum over $m$ and then over $n$ . Since $m=n+k$ and both $n$ and $k$ range from $0$ to $\infty$ , the sum over $m$ ranges from $0$ to $\infty$ . The new range of $n$ in the outer sum is determined by the conditions that $n\geq 0$ and $n=m-k$ , where $m$ and $k$ range from $0$ to $\infty$ . Consequently, $n$ has a lower limit of 0 and an upper limit of $m$ . Eq431, after swapping the order of summation, then becomes