# Padé Approximants of Power Series with Natural Boundaries

Consider a power series $$\sum_{n=0}^{\infty}c_{n}z^{n}$$ for which $$c_{n}\in\left\{ 0,1\right\}$$ for all $$n$$. One can write this as: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$ for some set $$V$$ of positive integers. I call this the “set-series” of $$V$$. There is a beautiful theorem due to Gábor Szegő which, for the case of set-series, shows that $$\varsigma_{V}\left(z\right)$$ is either a rational function whose poles are simple and located at roots of unity, or that $$\varsigma_{V}\left(z\right)$$ is a transcendental function with the unit circle ($$\partial\mathbb{D}$$) as a natural boundary.

Natural boundaries generally occur as the result of singularities clustering arbitrarily close to one another. My intuition tells me that in the case where $$\varsigma_{V}\left(z\right)$$ has a natural boundary (example: $$V=\left\{ 2^{n}:n\geq0\right\}$$, $$V=\left\{ n^{2}:n\geq0\right\}$$, etc), the clustering singularities in question are simple poles.

I figure a good way to try to see this would be via Padé approximants. The “rigorous” statement of my intution would then be something along the lines of: for an appropriately chosen sequence of Padé approximants $$\left\{ P_{n}\left(z\right)\right\} _{n\geq1}$$ of $$\varsigma_{V}\left(z\right)$$ (where $$\varsigma_{V}\left(z\right)$$ has a natural boundary on $$\partial\mathbb{D})$$, for every $$\epsilon>0$$ and every $$\xi\in\partial\mathbb{D}$$, there is an $$N_{\epsilon,\xi}\geq1$$ so that, for all $$n\geq N_{\epsilon,\xi}$$, any pole $$s$$ of $$P_{n}\left(z\right)$$ satisfying $$\left|s-\xi\right|<\epsilon$$ is necessarily simple.

With the literature on Padé Approximants appears to be quite extensive (while the literature on natural boundaries appears to be comparatively paltry), I was wondering if anyone knew of anything about this question, or something similar. Insight and/or references would be most appreciated.

Edit: having done some investigations, the above statements need to be modified. You can't have a dense set of simple poles on the boundary, otherwise you'd have an essential singularity. Thus, all but finitely many of the singularities of the natural boundary function on its natural boundary must be $$o\left(\frac{1}{z-\xi}\right)$$ as $$z$$ tends radially to $$\xi\in\partial\mathbb{D}$$.

• Oh my! I guess I've been getting his name wrong! It is Gabor Szëgo. The reference is page 260 of Reinhold Remmert's "ClassicalTopics in Complex Function Theory" (1998). The theorem states that for a power series centered at zero whose coefficients take on only finitely many distinct values, then either the unit disk is a natural boundary for the power series, or the power series defines a rational function whose poles are roots of unity. – MCS May 25 '19 at 19:52
• It's Gábor Szegő. – Ira Gessel May 26 '19 at 4:04
• I have fixed it. – MCS May 26 '19 at 20:17
• A minor point, but it's a long umlaut on the o: ő, not ö. – Ira Gessel May 27 '19 at 2:25
• The easiest way is to copy and paste. Search for "Gabor Szego" and the first hit is the Wikipedia page with the correct diacritics. (Or just copy from my comment.) – Ira Gessel May 28 '19 at 0:28