Continued Fractions and Irrationality Measures for Chowla--Selberg Gamma Quotients

Abstract

We give 39 rapidly convergent continued fractions for Chowla--Selberg gamma quotients, and deduce good irrationality measures for 20 of them, including for $\mathrm{CS}(-3)=(\Gamma (1/3)/\Gamma (2/3){)}^3$, for $a^{1/4}\mathrm{CS}(-4)=a^{1/4}(\Gamma (1/4)/\Gamma (3/4){)}^2$ with $a=12$ and $a=1/5$, and for $\mathrm{CS}(-7)=\Gamma (1/7)\Gamma (2/7)\Gamma (4/7)/(\Gamma (3/7)\Gamma (5/7)\Gamma (6/7))$. These appear to be the first proved and reasonable irrationality measures for gamma quotients.

27 pages