I'm struggling with the following problem:
In order to calculate $\int_{a}^{b}f(x)dx$ we use the Gaussian quadrature formula $\int_{a}^{b}f(x)dx\approx\sum_{i=0}^{n}A_if(x_i)$, where $A_i$ are the weights and $x_i$ are the roots of the polynomial $P_{n+1}(x)$ of degree $n+1$ with $P_0(x) = 1$ and $P_k(1) = 1 \forall k\geq 0$ which satisfies $\int_a^bx^kP_{n+1}(x)=0\forall k\in\{0,1,\ldots , n\}$.
Show that the Legendre polynomials $P_n$ satisfy Bonnet's recursion formula $(n+1)P_{n+1}(x) = (2n+1)xP_n(x)-nP_{n-1}(x)\forall n\geq 1$.
I've tried writing the polynomials in their canonical forms, but I have zero idea of how to relate the polynomials to each other.
It's also worth noting that I'm only allowed to use the conditions given in the problem, anything else has to be proven separately.
Any ideas on how I can continue(or start)?