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- TL;DR Summary
- I'm studying probability generating functions (PGF) and it is claimed that if two RVs have the same PGF, then they have the same distribution. The author of my book refers to the literature for this result and remarks this follows from the "uniqueness theorem" for power series. I have been trying to look up this theorem and found two theorems in regards to this, but to a great extent different proofs. This makes me think these theorems are not the same, but I can't tell the difference.

The first theorem is from baby Rudin, i.e. his book PMA. It reads as follows:

I omit the proof, as it is a bit technical. But basically one needs to prove that ##A##, the set of all limit points of ##E## in ##S##, is open, from which it'll follow that ##E=S##.

The second theorem is from these lecture notes:

The proof here is different from that in Rudin, much simpler it seems. But I don't know if maybe the corollary is weaker than the theorem in Rudin's book. Does anyone what the difference is between these two statements?

8.5 TheoremSuppose the series ##\sum a_n x^n## and ##\sum b_n x^n## converge in the segment ##S=(-R,R)##. Let ##E## be the set of all ##x\in S## at which $$\sum_{n=0}^\infty a_nx^n=\sum_{n=0}^\infty b_n x^n.$$ If ##E## has a limit point in ##S##, then ##a_n=b_n## for ##n=0,1,2,\ldots##. Hence the above equation holds for all ##x\in S##.

I omit the proof, as it is a bit technical. But basically one needs to prove that ##A##, the set of all limit points of ##E## in ##S##, is open, from which it'll follow that ##E=S##.

The second theorem is from these lecture notes:

Corollary 10.23.If two power series $$\sum_{n=0}^{\infty} a_n(x-c)^n, \quad \sum_{n=0}^{\infty} b_n(x-c)^n$$ have nonzero-radius of convergence and are equal in some neighborhood of ##c##, then ##a_n=b_n## for every ##n=0,1,2, \ldots##.

Proof.If the common sum in ##|x-c|<\delta## is ##f(x)##, we have $$a_n=\frac{f^{(n)}(c)}{n!}, \quad b_n=\frac{f^{(n)}(c)}{n!},$$ since the derivatives of ##f## at ##c## are determined by the values of ##f## in an arbitrarily small open interval about ##c##, so the coefficients are equal.

The proof here is different from that in Rudin, much simpler it seems. But I don't know if maybe the corollary is weaker than the theorem in Rudin's book. Does anyone what the difference is between these two statements?