Prove Bonnet's recursion formula for Legendre polynomials

Question

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)?

Answer 1

We may tackle this problem along the following lines. Assume for first $[a,b]=[-1,1]$. Up to translation and rescaling, this is not restrictive. Then we want a sequence of polynomials $\{P_n(x)\}_{n\in\mathbb{N}}$ with the property that $P_n(x)$ has degree $n$ and for any $k\in[0,n-1]$ $$ \int_{-1}^{1} x^k P_n(x)=0 $$ holds, so we may build the sequence $\{P_n(x)\}_{n\in\mathbb{N}}$ by applying the Gram-Schmidt process to the monomial base $\{1,x,x^2,\ldots\}$ of $L^2(-1,1)$ with the inner product $$ \langle f,g \rangle = \int_{-1}^{1} f(x)\,g(x)\,dx $$ and by choosing the normalization constraint: $$ \int_{-1}^{1} P_n(x)^2\,dx = \frac{2}{2n+1}.$$ In such a way, our sequence of polynomials is unique. In the next step, we take a sequence of polynomials defined by: $$ Q_n(x) = \frac{1}{2^n n!}\cdot\frac{d^n}{dx^n}\left[\left(x^2-1\right)^n\right] $$ and check, through integration by parts, that this sequence fulfills the same orthogonality properties of the previous sequence, hence $P_n(x)\equiv Q_n(x)$. The previous formula is known as Rodrigues' formula, and leads to the following generating function for Legendre polynomials: $$ \frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n\geq 0}P_n(x)\, t^n.$$ Once we have the generating function at our disposal, to prove the Bonnet's recursion formula is easy through a differentiation trick: just see the first lines of this Wikipedia page.