What are the Different Ways to Define Legendre Polynomials?

[JM00404]

Evening. I am having difficulty solving the the problems that have been included below. For the first problem, I essentially followed what the hint suggested that I do and I still cannot "see" the solution. I am honestly not sure how to even go about solving the second problem. For the third problem, I differentiated the generating function for the legendre polynomial, which gave a solution that was somewhat similar to the "Henyey-Greenstein phase function" but deviates from it slightly (the answer expression that I have right now is
$$2[r^2-1](\frac{d}{dx})[1+r^2-2cos(\theta)]^{-1/2}$$.
Finally, for the fourth problem, I really don't know where to begin. Any and all assistance/guidance/hints would be very much appreciated. Thank you.

Regards

I. Show that the nth Legendre polynomial is given by
$$P_n(x)=\sum_{k=0}^n\left(\stackrel{n}{k}\right)\left(\stackrel{n+k}{k}\right)\left(\stackrel{x-1}{2}\right)^k.$$
Hint: Write $$(x^2-1)^n=(x-1)^n[(x-1)+2]^n$$, apply the binomial expansion to the term in [\cdots], and differentiate n-times.

II. Find the degree three Legendre approximation of the function
$$f(x)=\left\{\stackrel{0 (-1\leq x<0)}{1 (0\leq x<1)}.$$

III. Use the formula
$$\frac{1}{\sqrt{1+r^2-2rx}}=\sum_{n=0}^\infty P_n(x)r^n$$
to derive the formula for the ``Henyey-Greenstein phase function''
$$\frac{1-r^2}{(1+r^2-2rcos(\theta))^{3/2}}=\sum_{n=0}^\infty (2n+1)P_n(cos(\theta))r^n$$.

IV. Let $$c_n$$ be the leading term of $$P_n$$ and set
$$\tilde{P}_n=c^{-1}P_n=x^n+$$ (lower powers of x).
Prove that if $$Q=x^n+\cdots$$ is any polynomial of degree n with leading coefficient
one, then
$$<Q,Q>\geq<\tilde{P}_n,\tilde{P}_n>$$
with equality only if $$Q=\tilde{P}_n$$.
(Hint: Write $$Q=\tilde{P}_n+h$$, where h is a linear
combination of $$P_0,P_1,\ldots P_{n-1}$$, and note that $$<Pn,h>=0$$.)

 

[HallsofIvy]

There are several equivalent ways of defining the Legendre polynomials. It is impossible to know how YOU should do these without know what definitions or formulas you have available to you.