Challenge of the week - Continued fractions for predictive modeling

Continued fractions is a fascinating subject, see picture below. They are extremely stable from a numerical point of view, have tons of useful properties, and are thus more robust against over-fitting, compared with standard linear regression. They have been extensively studied in the context of approximation and numerical analysis. Why is there no statistical theory of continued fractions? Why aren't these beautiful and powerful mathematical objects not used in data science?