Asymptotic behavior of integrals of Legendre polynomials

Question

By definition $\int_{-1}^1 |P_n(x)|^2 dx = O(n^{-1})$. What about the other powers? Do we know how $\int_{-1}^1 |P_n(x)|^k dx$ behaves for any $k$? Maybe $O(n^{-k/2})$?

Answer 1

Interesting question. If $P_n(x)$ is replaced with $T_n(x)$ (Chebyshev polynomial of the first kind) and $dx$ is replaced with $\frac{dx}{\sqrt{1-x^2}}$ we have

$$\begin{eqnarray*} \int_{-1}^{1}|T_n(x)|^k\frac{dx}{\sqrt{1-x^2}} = \int_{-\pi}^{\pi}|\cos(n\theta)|^k\,dt &=& \frac{1}{n}\int_{-n\pi}^{n\pi}|\cos\theta|^k\,d\theta\\&=&4\int_{0}^{\pi/2}\cos^k\theta\,d\theta\end{eqnarray*} $$ that does not depend on $n$ but on $k$ only, $O\left(\frac{1}{\sqrt{k}}\right)$. To deal with Legendre polynomials is slightly more difficult, since their roots do not have such a nice distribution, but in any case we may compute $$ \int_{-1}^{1}|P_n(x)|^{k}\,dx $$ for even values of $k$ through the recurrence relations of Legendre polynomials.

The Cauchy-Schwarz inequality gives an interpolation inequality: $$ \forall k,\qquad \left(\int_{-1}^{1}|P_n(x)|^{k}\,dx\right)\cdot\left(\int_{-1}^{1}|P_n(x)|^{k+2}\,dx\right)\geq \left(\int_{-1}^{1}|P_n(x)|^{k+1}\,dx\right)^2 $$ hence for a fixed $n$, the moment $\int_{-1}^{1}|P_n(x)|^{k}\,dx$ is a log-convex function of $k$, so a convex function.

We may tackle the original problem by exploiting a very strange property of the envelope of Legendre polynomials (thanks to David Speyer) that gives us the possibility to approximate $\int_{-1}^{1}|P_n(x)|^{k}\,dx$ with a value of the Euler beta function, since the zeroes of $P_n(x)$ behaves like the zeroes of $T_n(x)$. If $\theta\in(0,\pi)$ is not too close to the endpoints of such interval,

$$P_n(\cos \theta)\approx \frac{4}{\pi} \cdot \frac{2 \cdot 4 \cdots (2n)}{3\cdot 5 \cdots (2n+1)} \frac{\cos[(n+1/2) \theta - \pi/4]}{(2 \sin \theta)^{1/2}}$$

so the moments can be computed by using such formula plus the upper bound $$|P_n(x)|\leq\frac{\sqrt{2}}{\sqrt{\pi n}\sqrt[4]{1-x^2}}$$ near the endpoints of $[-1,1]$.

Notice that the dependency on $k$ is expected to be slightly irregular for small values of $k$, since the critical function $\frac{\sqrt{\frac{2}{\pi}}}{\sqrt[4]{1-x^2}}\in L^k(-1,1)$ only if $k<4$.