# $L^2$ norm of the first derivative of Legendre polynomials

Recently I have encountered a question while studying the orthogonal properties of Legendre's polynomial

$$\int_{-1}^{1} P_{n}^{\prime}(x) P_{n}^{\prime}(x)=n(n+1), n\geq1$$

I have tried the above question with the help of the generating function, but it doesn't help much.

## 1 Answer

From the recursions relations for the Legendre polynomials here, we have $$$$P'_n(x)=2\sum_{j=0}^{\lfloor\frac{n-1}{2}\rfloor} \frac{P_{n-1-2j}(x)}{\|P_{n-1-2j}\|^2}$$$$ for $$n\ge1$$, where $$\|P_{n}\|^2=\int_{-1}^1\left( P_n(x) \right)^2\,dx=\frac{2}{2n+1}$$ Then the integral can be written as \begin{align} I_n&=\int_{-1}^{1} P_{n}^{\prime}(x) P_{n}^{\prime}(x)\,dx\\ &=4\sum_{j=0}^{\lfloor\frac{n-1}{2}\rfloor}\sum_{k=0}^{\lfloor\frac{n-1}{2}\rfloor}\int_{-1}^{1}\frac{P_{n-1-2j}(x)}{\|P_{n-1-2j}\|^2}\frac{P_{n-1-2k}(x)}{\|P_{n-1-2k}\|^2}\,dx \end{align} Orthogonality of the polynomials imposes $$j=k$$ to obtain the non-vanishing terms in the summation. The integrals give the norm of remaining polynomials, then \begin{align} I_n&=4\sum_{j=0}^{\lfloor\frac{n-1}{2}\rfloor}\int_{-1}^{1}\frac{\left( P_{n-1-2j}(x) \right)^2}{\|P_{n-1-2j}\|^4}\,dx\\ &=4\sum_{j=0}^{\lfloor\frac{n-1}{2}\rfloor}\frac{1}{\|P_{n-1-2j}\|^2}\\ &=2\sum_{j=0}^{\lfloor\frac{n-1}{2}\rfloor}\left( 2n-4j-1 \right)\\ &=n(n+1) \end{align} independently of the parity of $$n$$.