Legendre poly, generating function

[physicsjock]

hey guys,

my lecturer skipped the proof to show that $\frac{1}{\sqrt{1+u^2 -2xu}}$ is a generating function of the polynomials,

he told us that we should do it as an exercise by first finding the binomial series of
$\frac{1}{\sqrt{1-s}}$ then insert s = -u² + 2xu

he then said to expand out sⁿ and group together all the uⁿ terms,

this is what I've been doing for a while and i haven't been able to see the end,

$\frac{1}{\sqrt{1-x}}=\sum\limits_{n=0}^{\infty }{\frac{\left( -\frac{1}{2} \right)\left( -\frac{3}{2} \right)...\left( \frac{1}{2}-n \right)}{n!}}{{(x)}^{n}}\,\,\,\,\,let\,\,x=2xu-{{u}^{2}}$

${{(u(-u+2x))}^{n}}={{(-1)}^{n}}{{u}^{n}}\left[ {{u}^{n}}-2x\frac{n!}{(n-1)!}{{u}^{n-1}}+4{{x}^{2}}\frac{n!}{(n-2)!}{{u}^{n-2}}+...+\frac{n!}{r!(n-r)!}{{(-1)}^{-r}}{{u}^{n-r}}{{(2x)}^{n}}+...+{{(-1)}^{-n}}{{(2x)}^{n}} \right]$


is my approach incorrect?

 

[mighty2000]

First of all, I would like to commend you for taking the initiative to work on this proof on your own. It shows that you are dedicated to understanding the material and willing to put in the effort to do so. However, I would like to offer some guidance on your approach to this problem.

The first step in proving that \frac{1}{\sqrt{1+u^2 -2xu}} is a generating function of the polynomials is to understand the concept of a generating function. A generating function is a power series that encodes information about a sequence of numbers. In this case, the polynomials are represented by the coefficients of the power series.

To prove that \frac{1}{\sqrt{1+u^2 -2xu}} is a generating function of the polynomials, we need to show that the coefficients of the power series match the coefficients of the polynomials. This is where your lecturer's suggestion to use the binomial series comes in. The binomial series is a specific type of power series that can help us find the coefficients of the power series we are working with.

Your approach of expanding out sn and grouping together all the un terms is a good start, but it may not lead to the desired result. Instead, try using the binomial series directly to find the coefficients of the power series. Then, substitute in the given values of s = -u2 + 2xu and see if the coefficients match with those of the polynomials.

I hope this helps clarify the steps you need to take in order to complete this proof. Keep working on it and don't hesitate to ask for help if you get stuck. Good luck!