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Proof and generalization of conjectures of Ramanujan Machine

Shuma Yamamoto

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 Petkovšek's algorithm. Especially, in the former way, we can get strong generalization of 31 conjectures.

Proof and generalization of conjectures of Ramanujan Machine

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 Petkovšek's algorithm. Especially, in the former way, we can get strong generalization of 31 conjectures.
Paper Structure (3 sections, 47 theorems, 169 equations)

This paper contains 3 sections, 47 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Proof and Generalization of the conjectures for $\pi^2$, $\log(2)$ and Catalan's constant
  3. Proofs of the conjectures for $\pi^2$, $\log(2)$ and special values of the zeta function by Petkovšek's algorithm

Key Result

Lemma 1.1

Assume that $A_n$ and $B_n$ satisfy Then,

Theorems & Definitions (93)

  • Lemma 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.2
  • ...and 83 more