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New proofs of two identities of Ramanujan

Patrick Morton

Abstract

A proof of several identities of Ramanujan involving theta functions of level $7$ is given which uses a specific modular function for $Γ_1(7)$ and Klein's projective representation of $PSL(2,7)$ into $PSL(3, \mathbb{C})$. Four identities of Berndt and Zhang are derived as algebraic corollaries of the main proof.

New proofs of two identities of Ramanujan

Abstract

A proof of several identities of Ramanujan involving theta functions of level is given which uses a specific modular function for and Klein's projective representation of into . Four identities of Berndt and Zhang are derived as algebraic corollaries of the main proof.
Paper Structure (4 sections, 12 theorems, 95 equations)

This paper contains 4 sections, 12 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Klein's projective representation
  3. Proof of Ramanujan's identity (\ref{['eqn:4']}).
  4. Identities of Berndt and Zhang.

Key Result

Theorem 1

With $u,v, w$ defined as in (eqn:1), we have

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 1
  • proof
  • Lemma 1
  • proof
  • Theorem 4
  • ...and 13 more