Consider the integral $$I_n^{(a,b)} = \int_{-1}^1 (1-x)^a\,(1+x)^b\, P_n(x)\, dx,$$ where $P_n(x)$ is the $n$-th Legendre polynomial. Here's a plot of $|I_n^{(50,20)}|$ for $n=0,\dots,70$:
(I just chose the $a,b$ arbitrarily, but the same phenomenon holds for pretty much any choice, and for other polynomials in the Jacobi family as well).
As $n$ ranges from $0$ to $a+b$ the norm of this integral seems to decay like $e^{-n^2}$ (or something faster than exponential anyways). I want to prove a bound to this effect.
What I have tried thus far:
- It's possible to write out the value of the integral in closed form, but it comes out as an alternating sum of huge combinatorial terms which end up cancelling to produce a small number. These seem to be notoriously difficult to analyze and extract meaningful bounds.
- Method of stationary phase / integration by parts. Since the orthogonal polynomials oscillate rapidly for large $n$, and $(1-x)^a(1+x)^b$ is slowly varying (in fact it looks like a bell curve), this seems like a natural approach. Using the trig approximation to the Legendre polynomials one can, eventually, bound the integral (assuming $a\le b$ by a sum of terms resembling $$\frac{1}{n^a} \left| \int_{-1}^1 \frac{d^a}{dx^a}\left[(1+\cos x)^a (1-\cos x)^b\right] \sin x\,e^{i n x}\, dx \right|.$$ The problem here is that now the high order derivatives themselves oscillate quite substantially, and though they can be written down explicitly they contain large alternating terms that are hard to bound. In any case, even if this worked, the bound would only be polynomial in $n$.
- Recursion. By manipulating the integrand and using recurrence properties for the Legendre polynomials you can write an equation $I_n^{(a,b)}$ involving (two) other terms in the sequence. Since the boundary terms are relatively easy to compute, you can try to get a bound by applying the recursion. Here again though, the recursions rely on a lot of cancellations not to blow up, so the bound ends up being very bad.
Basically, I've tried everything I can think of and keep getting thwarted by these annoying cancellation problems. I guess one other technique to try would be steepest descent, but I have little experience with choosing the appropriate contour and suspect I might run into the same issue. Any tips from the experts would be most welcome. Thanks!