On Fabry’s Quotient Theorem

Abstract

We present a short proof of the Fabry quotient theorem, which states that for a complex power series with unit radius of convergence, if the quotient of its consecutive coefficients tends to s, then the point $z=s$ is a singular point of the series. This proof only uses material from undergraduate university studies.

MSC:

• Primary 30B10
• Secondary 30B40
• 42A32

Acknowledgment

I thank A. Borichev, A. Eremenko, D. Khavinson, F. Nazarov, and M. Sodin for valuable feedback. I especially thank F. Nazarov for an important suggestion that led to an improvement of the actual result we show (see the quantitative statement at the beginning of our proof of Theorem 1 in Section 2) and to simplification of our original arguments, and M. Sodin for introducing me to the Fabry theorems, encouraging me to write this article, and for his generous help with the presentation. I also thank the editor and the reviewers for careful reading of the article, and for valuable comments that significantly improved the presentation. This work was partially supported by ERC Starting Grant 757585 and ISF Grant 2026/17.

Additional information

Funding

This work was partially supported by ERC Starting Grant 757585 and ISF Grant 2026/17.

Notes on contributors

Lev Buhovsky

Lev Buhovsky is a professor at Tel Aviv University. He received his Ph.D. in Mathematics from Tel Aviv University, and later completed postdocs at MSRI and the University of Chicago. His research interests are symplectic geometry and analysis.