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my lecturer skipped the proof to show that [itex]\frac{1}{\sqrt{1+u^2 -2xu}}[/itex] 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

[itex]\frac{1}{\sqrt{1-s}}[/itex] then insert s = -u^{2}+ 2xu

he then said to expand out s^{n}and group together all the u^{n}terms,

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

[itex]\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}}[/itex]

[itex]

{{(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]

[/itex]

is my approach incorrect?

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# Homework Help: Legendre poly, generating function

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