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I am trying to solve identity involving binomials and Fibonacci numbers by using generating functions: $$\sum_{k=0}^n{n \choose k}{n+k\choose k}f_{k+1}=\sum_{k=0}^n{n \choose k}{n+k\choose k}(-1)^{n-k}f_{2k+1}$$

My computational approach by Mathematica lead me to derive this generating function:

$$\frac{\sqrt{3x^2-2x+3+2\sqrt{x^4-8x^3-2x^2-8x+1}}}{\sqrt{5}\sqrt{x^4-8x^3-2x^2-8x+1}}$$

Can someone show how to transform both or any of the identity sides to obtain (coefficiens of) this generating function.

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    I suspect you need to use the shifted Legendre polynomials here $$P_{n}(2x-1) =(-1)^{n}\sum_{k=0}^n{n \choose k}{n+k\choose k}(-x)^{k}.$$2011-07-26
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    :I meant to add that their generating function (the shifted Legendre polynomials) is immediately obtainable from the generating function for the Legendre polynomials (see e.g. the relevant Wikipedia entry). The other ingredient you need will be Binet's formula for the Fibonacci numbers.2011-07-26
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    Are you trying to *prove* the identity using generating functions? Or do you want to compute the coefficients?2014-01-07

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