# Convergence of expansion of Legendre generating function.

 P: 86 The Legendre functions may be defined in terms of a generating function: $$g(x,t) = \frac{1}{\sqrt{1-2xt+t^2}}$$ Of course, $$\frac{1}{\sqrt{1+x}} =\sum^{\infty}_{n=0} (\stackrel{-.5}{n})x^n$$. However, this series doesn't converge for all x. It only converges if |x| < 1. In our case, $$|t^2 - 2xt|$$ 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.