Convergence of expansion of Legendre generating function.

[scorpion990]

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.

 

[gammamcc]

The interval of convergence can depend on the point at which you are taking the
Taylor expansion - and, for that matter, which variable you are doing the
expansion in. The one you choose above is just one of many ways to expand it, but not
the way the Legendre functions show up.
The Legendre functions (http://en.wikipedia.org/wiki/Legendre_polynomials) would
come from an expansion in your variable t. The assumption for certain applications would
be that |x| < or =1 , |t| < 1 , which does not check for your series (eg. x=-1, t=.999999999). [ I don't recall off-hand how the series extends for |t|> or + 1 .]

As an expansion in t (|t|<1) , you can quickly verify the region of convergence (going to the complex plane) by using Cauchy's integral formula for upper bounds on the Taylor coefficients followed by a series test.