The Legendre functions may be defined in terms of a generating function: [tex]g(x,t) = \frac{1}{\sqrt{1-2xt+t^2}} [/tex](adsbygoogle = window.adsbygoogle || []).push({});

Of course, [tex]\frac{1}{\sqrt{1+x}} =\sum^{\infty}_{n=0} (\stackrel{-.5}{n})x^n [/tex].

However, this series doesn't converge for all x. It only converges if |x| < 1. In our case, [tex]|t^2 - 2xt|[/tex] would have to be less than 1.

In the derivation of many recursion formulas, powers of t are set equal to each other. However, this isn't valid for all values of t and x... How come this method of derivation is still valid? Any help/insight would be appreciated.

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# Convergence of expansion of Legendre generating function.

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