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I am trying to understand how can we generate the Legendre Polynomials $P2(x)$ and $P3(x)$ by using the Bonnet’s recursion formula which I know that is :

$(n+1)Pn+1(x)=(2n+1)xPn(x)-nPn-1(x)$

and given $P0(x) = 1$ and $ P1(x)=x$

I found the first few Legendre polynomials in https://en.wikipedia.org/wiki/Legendre_polynomials

So my question is if we use the Bonnet’s recursion formula we should obtain the Legendre polynomials mentioned in the above link? How can we do that?

Can anyone give me a hint or a place where I can find more information and examples?

Thank you

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    Bonnet's recursion is equivalent to Rodrigues' formula, giving Legendre polynomials in terms of the derivatives of $(1-x^2)^n$. Rodrigues' formula is more practical to use if you need to extract a specific coefficient, for instance. – 2017-02-06
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    Thank you for your help. I will now study and use Rodrigues' formula. – 2017-02-07

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