Hello!

Working on the binomial series I found (empirically) this equality:

Attachment 4881

Can someone help and demonstrate it formally for me?

I tried unsuccesfully… :(

Thanks in advance!

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- Jan 19th 2008, 02:15 PMpaolopiaceBinomial coefficients
Hello!

Working on the binomial series I found (empirically) this equality:

Attachment 4881

Can someone help and demonstrate it formally for me?

I tried unsuccesfully… :(

Thanks in advance! - Jan 19th 2008, 05:02 PMtopsquark
- Jan 19th 2008, 09:24 PMmr fantastic
You can do it in a sloppy way .....

$\displaystyle {-1 \choose k} = \frac{(-1)!}{k!(-1 - r)!}$

Apply the factorial dogmatically:

$\displaystyle = \frac{(-1)(-2)(-3) .... (-k)(-k-1)!}{(-1 - k)! k!}$

Cancel the bad stuff:

$\displaystyle = \frac{(-1)^k k!}{k!} = ....$

And you're left with the good stuff.

I imagine a formal approach would use the gamma function (with great care since $\displaystyle x = -n$, $\displaystyle n \in \mathbb{Z}$, are simple poles .....) Maybe if someone has the time and the inclination (but I don't think a pre-algebra and algebra forum is the place) ..... - Jan 19th 2008, 10:42 PMpaolopiaceTo mr fantastic...
That's the way I did prior to posting here.

I'd really like a more formal proof...