# gamma function algebra...

• October 18th 2007, 10:13 PM
PiePie
gamma function algebra...
Wasn't sure which forum to put this in, I've done the probability part - it's just that my answer is a few lines away from the sample solution I've been given, and I don't understand what they've done.

$(2^k/sqrt(pi))(k - 1/2)(k - 3/2) ... ... (3/2)(1/2)Gamma(1/2)$

The next line of working for the sample solution is

$(2k - 1)(2k - 3) ... ... (3)$

I see that the sqrt(pi) and the gamma(1/2) cancel, other than that I'm stuck! :(

Any help would be much appreciated. not sure if it's needed or not, but k = n/2 where n is an positive even integer
• October 19th 2007, 05:53 AM
topsquark
Quote:

Originally Posted by PiePie
Wasn't sure which forum to put this in, I've done the probability part - it's just that my answer is a few lines away from the sample solution I've been given, and I don't understand what they've done.

$(2^k/sqrt(pi))(k - 1/2)(k - 3/2) ... ... (3/2)(1/2)Gamma(1/2)$

The next line of working for the sample solution is

$(2k - 1)(2k - 3) ... ... (3)$

I see that the sqrt(pi) and the gamma(1/2) cancel, other than that I'm stuck! :(

Any help would be much appreciated. not sure if it's needed or not, but k = n/2 where n is an positive even integer

Unless you have a specific value for k the best you can do is
$(2k - 1)(2k - 3)~ ... ~3 \cdot 1 = (2k - 1)!!$
where (2k - 1)!! is the "odd factorial function." It multiplies only the odd numbers up to (2k - 1).

(In case you were wondering there is an even factorial function that is a product of only even numbers, and it has the same symbol, "!!" The nice thing about this is you don't really need to use it:
$(2n)!! = 2^n \cdot n!$)

-Dan
• October 19th 2007, 03:25 PM
PiePie
Thanks for the reply... I actually meant I don't understand how to go from the first line I posted to the second. Sorry for the confusion...
I think I just don't understand what happens to the "k" index?