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Math Help - Proof of Pfaff's reflection law... There is a clue but I still don't know how to.

  1. #1
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    Proof of Pfaff's reflection law... There is a clue but I still don't know how to.

    I need to proof Pfaff's law:



    It's supposed to be proven 'by comparing the coefficients of z^n on both sides of the equation'.
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  2. #2
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    I don't know much about hypergeometric functions, but according to this article (pdf file), F\left({a,b\atop c}\,\Big|\,z\right) is given by the integral

    F\left({a,b\atop c}\,\Big|\,z\right) = \frac1{\mathrm{B}(b,c-b)}\int_0^1t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,dt,

    where the B denotes the beta function.

    If you are willing to accept that formula, then the relation F\left({a,c-b\atop c}\,\Big|\,z\right) = \frac1{(1-z)^a}F\left({a,b\atop c}\,\Big|\,\frac{-z}{1-z}\right) follows very easily by making the substitution t\mapsto1-t in the integral.
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  3. #3
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    Thanks for your help! I'm trying to work it out right now, I hope I will understand it...
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