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To prove the recurrence relation for the Legendre polynomials \( L_n(x) \):

\[ L_{n+1}(x) = (2n+1-x)L_{n}(x) - n^2 L_{n-1}(x) \]

we can start with the generating function of Legendre polynomials. The generating function is given by:

\[ \frac{1}{\sqrt{1 - 2xt + t^2}} = \sum_{n=0}^{\infty} L_n(x)t^n \]

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