fraction in combination...please tell me the calculation


by maverick6664
Tags: calculation, combinationplease, fraction
maverick6664
maverick6664 is offline
#1
Dec24-05, 07:56 AM
P: 80
Hi,


I'm reading the proof of Rodriguez recurrence formula
[tex]P_l(x) = \frac{1}{2^l l!} \frac{d^l}{dx^l} (x^2-1)^l[/tex]

This formula itself isn't a problem.

But during the proof I got
[tex](1-2xt+t^2)^{-\frac{1}{2}} = \sum_n \left( \begin{array}{c} -\frac{1}{2} \\ n \end{array} \right) (-2xt)^n(1+t^2)^{-(\frac{1}{2})-n} [/tex]
and wondering what the fraction in [tex]\left( \begin{array}{c} -\frac{1}{2} \\ n \end{array} \right)[/tex] means (and that it's negative)... and I don't know the range of [tex]n[/tex] in this summation (maybe 0 to indefinate?). Actually if this fraction is allowed, this formula makes sense.

Will anyone show me the definition of this kind of combination? Online reference will be good as well.

Thanks in advance! and Merry Christmas!
Phys.Org News Partner Mathematics news on Phys.org
Hyperbolic homogeneous polynomials, oh my!
Researchers help Boston Marathon organizers plan for 2014 race
'Math detective' analyzes odds for suspicious lottery wins
mathman
mathman is offline
#2
Dec24-05, 08:24 PM
Sci Advisor
P: 5,942
The sum (n) goes from 0 to infinity. The coefficient you are looking at is the generalization of the binomial coefficient.
The first few terms are 1, -1/2, (-1/2)(-3/2)/2!, (-1/2)(-3/2)(-5/2)/3!.
If you look at a binomial expansion of (a+b)c,
you have a=1+t2, b=-2xt and c=-1/2.
maverick6664
maverick6664 is offline
#3
Dec24-05, 10:17 PM
P: 80
Thanks! It helps me a LOT. The keywords are what I needed

I will learn Pochhammer symbol.

EDIT: oh...thinking of Taylor expansion, proof is easy, but I've never seen that form of binomial expantion


Register to reply

Related Discussions
Combination Probability General Math 2
Combination Precalculus Mathematics Homework 1
Combination Please Help Introductory Physics Homework 30
combination Precalculus Mathematics Homework 3
how many possible arrangements Permutations Set Theory, Logic, Probability, Statistics 1