Why is the Gamma function undefined for non-positive integers?

[roger]

what is nCr iff r>n?

what does it mean to choose say 5 objects from 4?

My understanding was that nCr is defined where n>r but I have a question involving this which I don't understand.

thanks
Roger

 

[chroot]

By the defintion of the binomial coefficient "nCr," you'll get a number smaller than one when r > n. It's still "defined," but it doesn't make much sense to say you can choose 5 objects from 4 in 0.2 different ways.

- Warren

 

[mathman]

nCr=n!/(r!(n-r)!). If r>n, the denominator is oo, so nCr=0.

 

[chroot]

Actually, mathman, you're right. I forgot that the subtraction on the bottom will result in a negative number... but the factorial of a negative number is undefined, and thus so is nCr where r > n.

- Warren

 

[roger]

what do you mean warren? mathman didnt say that he said If r>n, the denominator is oo, so nCr=0.

 

[chroot]

If r > n, then the "(n-r)!" in the denominator is undefined. The factorial of negative numbers is undefined.

- Warren

 

[roger]

so why did he say it is equal to zero? and why did he say the denominator is oo?

 

[chroot]

so why did he say it is equal to zero? and why did he say the denominator is oo?

I don't know. Maybe you should ask him.

- Warren

 

[roger]

well it can't be undefined and zero at the same time it must be one or the other and if it is undefined I don't know why it is mentioned in the question I have.

 

[John Creighto]

I don't know much about that function. I wonder if they use the gamma function for non natural numbers.

http://en.wikipedia.org/wiki/Gamma_function

 

[chroot]

The gamma function is also undefined for negative real numbers. No matter how you slice it, "4 C 5" is undefined.

- Warren

 

[John Creighto]

The gamma function is also undefined for negative real numbers. No matter how you slice it, "4 C 5" is undefined.

- Warren

Are you sure about that? The graph of the gamma function on wikipedia would suggest otherwise to me. Maybe I am misreading it.

 

[d_leet]

The gamma function is also undefined for negative real numbers. No matter how you slice it, "4 C 5" is undefined.

- Warren

Negative inegers, I'm pretty sure the gamma function is defined for other negative numbers, just not the negative integers.

 

[Mute]

According to the wikipedia page on the binomial coefficient, the coefficient is defined to be zero for n < r (or r < 0 - there is a generalization to negative integers, though.).

http://en.wikipedia.org/wiki/Binomial_coefficient#Definition

Since this is just the definition of the binomial coefficient, not necessarily as it applies to combinatorics, one might argue that it doesn't apply to counting problems... However, it makes sense to me that there are zero ways to choose 5 objects from a set of 4, for instance.

 

[EnumaElish]

Mathematica defines nCm as Binomial[n,m] = Γ(n+1)/(Γ(m+1)Γ(n-m+1)), seemingly = 0 when m > n.

If one defines nCm as combination with repetition, then the formula becomes (n+m-1)!/(m!(n-1)!), which does allow m > n.

 

[mathman]

(-1)!*0=0!=1, therefore (-1)!=1/0. 1/0 is not finite. For other negative integers, you will have a similar result.

 

[chroot]

-1! is undefined, not infinite. That's what your equation indicates, too.

- Warren

 

[roger]

I'm getting varied answers and abit confused at that. What's the bottom line?

 

[mathman]

Look up Gamma function on Wikepedia (Gamma(n+1)=n!). You will see a graph as a function of real x. At 0 and each of the negative integers you will see a vertical line asymptote. On one side of each asymptote, the curve goes to +oo, on the other the curve goes to -oo. In that sense it is undefined (-oo or +oo), but certainly not finite. Therefore nCr for r>n is always 0.

 

[EnumaElish]

I'm getting varied answers and abit confused at that. What's the bottom line?

If r > n, nCr = 0 without repetition but nCr > 0 with repetition.

 

[chroot]

Look up Gamma function on Wikepedia (Gamma(n+1)=n!). You will see a graph as a function of real x. At 0 and each of the negative integers you will see a vertical line asymptote. On one side of each asymptote, the curve goes to +oo, on the other the curve goes to -oo. In that sense it is undefined (-oo or +oo), but certainly not finite. Therefore nCr for r>n is always 0.

(1 / undef) is undef. It's certainly not zero.

- Warren

 

[uart]

(1 / undef) is undef. It's certainly not zero.

- Warren

$$x^{-1}$$ is undefined at x=0 but $$1/x^{-1}$$ is well defined and certainly zero. Right ot wrong I'm picturing mathmans argument as something like this. In any case I agree that nCr is zero for r > n.

 

[CRGreathouse]

Negative inegers, I'm pretty sure the gamma function is defined for other negative numbers, just not the negative integers.

Yes, the gamma function is defined on all real numbers except the nonpositive integers:
$$\mathbb{R}\setminus\{0,-1,-2,-3,\ldots\}$$

 

[uart]

Yes we all agree that the Gamma function is undefined for non-positive integers. The reciprocal 1/Gamma however approaches zero from both the left and the right at each non-positive integer.