Fraction in combination please tell me the calculation

In summary, the conversation is about the proof of the Rodriguez recurrence formula for P_l(x), which is not a problem itself. During the proof, the speaker encountered a summation involving a negative fraction, which is the generalization of the binomial coefficient. The sum goes from 0 to infinity and the speaker plans to learn more about the Pochhammer symbol.
  • #1
maverick6664
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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!
 
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  • #2
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.
 
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  • #3
Thanks! It helps me a LOT. The keywords are what I needed :smile:

I will learn Pochhammer symbol.

EDIT: oh...thinking of Taylor expansion, proof is easy, but I've never seen that form of binomial expansion :frown:
 
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1. What is a fraction?

A fraction is a mathematical expression representing a part of a whole. It is written in the form of a numerator (the top number) over a denominator (the bottom number). For example, in the fraction 3/4, 3 is the numerator and 4 is the denominator.

2. How do you add fractions?

To add fractions, you need to have a common denominator. If the fractions already have a common denominator, you can simply add the numerators and keep the denominator the same. If the fractions have different denominators, you need to find the least common multiple (LCM) of the denominators and convert the fractions to have that as the new denominator. Then, you can add the numerators and keep the denominator the same.

3. How do you subtract fractions?

To subtract fractions, you also need a common denominator. If the fractions already have a common denominator, you can simply subtract the numerators and keep the denominator the same. If the fractions have different denominators, you need to find the LCM of the denominators and convert the fractions to have that as the new denominator. Then, you can subtract the numerators and keep the denominator the same.

4. How do you multiply fractions?

To multiply fractions, you simply multiply the numerators and multiply the denominators. The resulting fraction may need to be simplified by finding the greatest common factor (GCF) of the numerator and denominator and dividing both by the GCF.

5. How do you divide fractions?

To divide fractions, you flip the second fraction (the divisor) and multiply it by the first fraction (the dividend). The resulting fraction may need to be simplified by finding the GCF of the numerator and denominator and dividing both by the GCF.

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