I would like to ask for some help regarding a problem with Legendre polynomials:
P_n(x) is an arbitrary Legendre polynomial (0<=n and -1<x<1). Prove the following
Thank you in advance!
Well, I don't think that induction is the right choice, because my professor said that I should start from the Rodrigues-formula (Rodrigues' formula - Wikipedia, the free encyclopedia) and than use some complex Cauchy-integrals. However, my efforts weren't successful yet.
So, I hope someone can give me some ideas how to approach this problem.
Alright, let's see. Here is what would have happened if you tried induction specifically. Suppose that up to and including some , for some function . We have the Bonnet recursion formula
which you can derive from the other formula. Now
all obtained by the triangle inequality and the bounds on . So,
So, technically, you can do an induction but the function you bound with will depend on . I'll take a look at this complex integrals approach and let you know.
Rearrange the formula for to get
The last one we can obtain by the -th Cauchy integral formula
where is a circle with radius, say , centered at the origin. So, we have obtained an integral formula for the Legendre polynomials
Now it might just be an exercise in estimating this integral
I really appreciate your help Vlasev.
Unfortunately, I think that I am stuck again. I tried to estimate a absolute value of the integral, but what I get that isn't close to the original upper limit. I tried using the fact the absolute value of an integral is smaller than the integral of an absolute value.
The results somehow seems to depend on the value of , however I don't understand why should it. Could you please give me some more help?