# generating function of Pn(x)

• Apr 16th 2012, 12:04 AM
Jesssa
generating function of Pn(x)
hey guys,

my lecturer skipped the proof to show that $\displaystyle \frac{1}{\sqrt{1+u^2 -2xu}}$ 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
$\displaystyle \frac{1}{\sqrt{1-s}}$ 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,

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

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

is my approach incorrect?