On the Connection Between Irrationality Measures and Polynomial Continued Fractions

Abstract

Linear recursions with integer coefficients, such as the one generating the Fibonacci sequence, have been intensely studied over millennia and yet still hide new mathematics. Such a recursion was used by Ap\'ery in his proof of the irrationality of $\zeta(3)$, later named the Ap\'ery constant. Ap\'ery's proof used a specific linear recursion containing integer polynomials forming a continued fraction; called polynomial continued fractions (PCFs). Similar polynomial recursions prove the irrationality of other mathematical constants such as $\pi$ and $e$. More generally, the sequences generated by PCFs form Diophantine approximations (DAs), which are ubiquitous in areas of math such as number theory. It is not known which polynomial recursions create useful DAs and whether they prove irrationality. Here, we present general conclusions and conjectures about DAs created from PCFs. Specifically, we generalize Ap\'ery's work, going beyond his particular choice of PCF, finding the conditions under which a PCF proves irrationality or provides an efficient DA. To provide concrete examples, we apply our findings to PCFs found by the Ramanujan Machine algorithms to represent fundamental constants such as $\pi$, $e$, $\zeta(3)$, and the Catalan constant G. For each such PCF, we demonstrate the extraction of its convergence rate and efficiency, as well as the bound it provides for the irrationality measure of the fundamental constant. We further propose new DA conjectures based on PCFs. Our findings motivate future research on sequences created by any linear recursions with integer coefficients, to aid the development of systematic algorithms for finding DAs of fundamental constants. Consequently, our study may contribute to ongoing efforts to answer open questions, such as the proof of the irrationality of the Catalan constant or of values of the Riemann zeta function (e.g., $\zeta(5)$).