question on indices of a generating function

I want to find the generating function for a_{n}=n(n-1).

$\displaystyle \sum_{n=0}^{\infty }n(n-1)\frac{x^n}{n!}$

$\displaystyle =\sum_{n=0}^{\infty }\frac{x^n}{(n-2)!}$

$\displaystyle =x^2\sum_{n=0}^{\infty }\frac{x^{n-2}}{(n-2)!}$

$\displaystyle =x^2\sum_{k=-2}^{\infty }\frac{x^{k}}{k!}$

$\displaystyle =x^2\sum_{k=0}^{\infty }\frac{x^{k}}{k!}$

$\displaystyle =x^2e^x$

Am I justified in the second to last step, ignoring the summands for k=-1 and k=-2?

If so, is there a reason why I can ignore the first two terms?

Re: question on indices of a generating function

Is there such a thing as (-2)! or (-1)!?

Re: question on indices of a generating function

No, but I wasn't sure if (-2)! and (-1)! not existing allowed me to ignore them.

So because (-2)! and (-1)! don't exist, we can't evaluate them, hence it's okay to ignore them?

Re: question on indices of a generating function

Only in this case, because you can see if you plug n=0 and n=1 into your original sum that those terms are 0 anyway.

Really what you should have written is

$\displaystyle \displaystyle \begin{align*} \sum_{n = 0}^{\infty} n(n-1)\frac{x^n}{n!} &= 0 + 0 + \sum_{n = 2}^{\infty} n(n-1)\frac{x^n}{n!} \\ &= \sum_{n = 2}^{\infty} n(n-1)\frac{x^n}{n!} \\ &= \sum_{n = 2}^{\infty} \frac{x^n}{(n-2)!} \\ &= x^2 \sum_{n=2}^{\infty} \frac{x^{n-2}}{(n-2)!} \\ &= x^2\sum_{k=0}^{\infty}\frac{x^k}{k!} \\ &= x^2\,e^x \end{align*}$

It doesn't make sense to put into your counter values which will give undefined terms.

Re: question on indices of a generating function

Thank you very much, it now makes a lot of sense.