On the Connection Between Irrationality Measures and Polynomial Continued Fractions (2024)

Abstract Linear recursions with integer coefficients, such as the recursion that generates the Fibonacci sequence $${F}_{n}={F}_{n-1}+{F}_{n-2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , have been intensely studied over millennia and yet still hide interesting undiscovered mathematics. Such a recursion was used by Apéry in his proof of the irrationality of $$\zeta \left(3\right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ζ</mml:mi> <mml:mfenced> <mml:mn>3</mml:mn> </mml:mfenced> </mml:mrow> </mml:math> , which was later named the Apéry constant. Apéry’s proof used a specific linear recursion that contained integer polynomials (polynomially recursive) and formed a continued fraction; such formulas are called polynomial continued fractions (PCFs). Similar polynomial recursions can be used to prove the irrationality of other fundamental constants such as $$\pi$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> and $$e$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>e</mml:mi> </mml:math> . More generally, the sequences generated by polynomial recursions form Diophantine approximations, which are ubiquitous in different areas of mathematics such as number theory and combinatorics. However, in general it is not known which polynomial recursions create useful Diophantine approximations and under what conditions they can be used to prove irrationality. Here, we present general conclusions and conjectures about Diophantine approximations created from polynomial recursions. Specifically, we generalize Apéry’s work from his particular choice of PCF to any general PCF, finding the conditions under which a PCF can be used to prove irrationality or to provide an efficient Diophantine approximation. To provide concrete examples, we apply our findings to PCFs found by the Ramanujan Machine algorithms to represent fundamental constants such as $$\pi$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> , $$e$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>e</mml:mi> </mml:math> , $$\zeta \left(3\right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ζ</mml:mi> <mml:mfenced> <mml:mn>3</mml:mn> </mml:mfenced> </mml:mrow> </mml:math> , and the Catalan constant. 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 conjectures about Diophantine approximations based on PCFs. Looking forward, our findings could motivate a search for a wider theory on sequences created by any linear recursions with integer coefficients. Such results can help the development of systematic algorithms for finding Diophantine approximations 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., $$\zeta \left(5\right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ζ</mml:mi> <mml:mfenced> <mml:mn>5</mml:mn> </mml:mfenced> </mml:mrow> </mml:math> ).