Proof and generalization of conjectures of Ramanujan Machine

TL;DR

This paper proves 38 conjectures from the Ramanujan Machine project by solving associated difference equations, offering strong generalizations for most, and advances understanding of continued fractions for mathematical constants.

Contribution

It provides rigorous proofs for Ramanujan Machine conjectures and introduces methods for their generalization using differential equations and Petkovšek's algorithm.

Findings

38 conjectures proved

Strong generalizations obtained for 31 conjectures

Enhanced methods for analyzing continued fractions

Abstract

The Ramanujan Machine project predicts new continued fraction representations of numbers expressed by important mathematical constants. Generally, the value of a continued fraction is found by reducing it to a second order linear difference equation. In this paper, we prove 38 conjectures by solving the equation in two ways, use of a differential equation or application of Petkov\v{s}ek's algorithm. Especially, in the former way, we can get strong generalization of 31 conjectures.