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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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# Fraction in combination please tell me the calculation

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