1. ## Binomial coefficients

Hello!

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

Can someone help and demonstrate it formally for me?

I tried unsuccesfully…

2. Originally Posted by paolopiace
Hello!

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

Can someone help and demonstrate it formally for me?

I tried unsuccesfully…

Okay, I'll bite. Is this supposed to be the combinitorial function? Typically
${n \choose r} = \frac{n!}{r!(n - r)!}$

How can you define this for a negative n? Or does your symbol mean something else?

-Dan

3. Originally Posted by topsquark
Okay, I'll bite. Is this supposed to be the combinitorial function? Typically
${n \choose r} = \frac{n!}{r!(n - r)!}$

How can you define this for a negative n? Or does your symbol mean something else?

-Dan
You can do it in a sloppy way .....

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

Apply the factorial dogmatically:

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

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

And you're left with the good stuff.

I imagine a formal approach would use the gamma function (with great care since $x = -n$, $n \in \mathbb{Z}$, 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) .....

4. ## To mr fantastic...

That's the way I did prior to posting here.

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