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Series for $1/π$ arising from Cauchy product

Roman Le Lan

Abstract

In this note, we evaluate a series for $1/π$ conjectured by Sun. Our proof uses the Cauchy product and hypergeometric transformations. From this result, we derive two additional analogous series for $1/π$ involving polynomials of degree $3$. Further identities can be proved using our method; these are presented in a table at the end of the note.

Series for $1/π$ arising from Cauchy product

Abstract

In this note, we evaluate a series for conjectured by Sun. Our proof uses the Cauchy product and hypergeometric transformations. From this result, we derive two additional analogous series for involving polynomials of degree . Further identities can be proved using our method; these are presented in a table at the end of the note.

Paper Structure

This paper contains 4 sections, 4 theorems, 21 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Proof of Theorem \ref{['1']}
  4. Proof of Theorem \ref{['2']}

Key Result

Theorem 1

We have the following identity. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (8)

  • Theorem 1
  • Conjecture 1: Z.-W. Sun
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2: J. Guillera
  • proof : Proof of Theorem \ref{['1']}
  • proof : Proof of Theorem \ref{['2']}