Calculate Binomial Coefficient: {-3 \choose 0}, {-3 \choose 1}, {-3 \choose 2}

[LagrangeEuler]

Homework Statement

Calculate
$${-3 \choose 0}$$, $${-3 \choose 1}$$, $${-3 \choose 2}$$

Homework Equations

In case of integer $n$ and $k$
$${ n \choose k}=\frac{n!}{k!(n-k)!}=\frac{n(n-1)(n-2)...(n-k+1)}{k!}$$

The Attempt at a Solution

I am not sure how to calculate this. Any idea?[/B]

 

[haruspex]

I've never worked with such things, but I believe binomial coefficients can be extended to noninteger and negative arguments by using the gamma function instead of factorials. Google it.

 

[LagrangeEuler]

I've never worked with such things, but I believe binomial coefficients can be extended to noninteger and negative arguments by using the gamma function instead of factorials. Google it.

Yes but gamma function diverge for integer negative values. So $\Gamma(-3)=(-4)!=\infty$

 

[haruspex]

Yes but gamma function diverge for integer negative values. So $\Gamma(-3)=(-4)!=\infty$

Can you avoid that by some cancellation? You will have negative arguments in gamma functions both in the numerator and the denominator.

 

[hilbert2]

The limit $\lim_{x\to n}\frac{\Gamma(x+1)} {\Gamma(k+1)\Gamma(x-k+1)}$ can exist even if $n$ is negative. Try it with Wolfram Alpha.

 

[LagrangeEuler]

Not sure. For example ${-3 \choose 0}=\frac{(-3)!}{0!(-3-0)!}=1$. This is OK. But in case ${-3 \choose 1}=\frac{(-3)!}{(-4)!}=\frac{\Gamma(-2)}{\Gamma(-3)}=\frac{-3\Gamma(-3)}{\Gamma(-3)}=-3$
I think this is OK. Tnx :)

 

[Ray Vickson]

Homework Statement

Calculate
$${-3 \choose 0}$$, $${-3 \choose 1}$$, $${-3 \choose 2}$$

Homework Equations

In case of integer $n$ and $k$
$${ n \choose k}=\frac{n!}{k!(n-k)!}=\frac{n(n-1)(n-2)...(n-k+1)}{k!}$$

The Attempt at a Solution

I am not sure how to calculate this. Any idea?[/B]

The standard _definition_ of ${n \choose k}$ for non-negative integer $k$ and general real $n$ is given by the second formula you wrote in Post #1 (i.e., the formula that does not involve $n!$ or $(n-k)!$). Sometimes that formula can be expressed in terms of Gamma functions, and sometimes not.

Using that definition we obtain
$${-m \choose k} = (-1)^k {m+k-1 \choose k}$$
for integers $m,k \geq 0$.