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On Eisenstein series identities and new identities connecting Ramanujan-Göllnitz-Gordon continued fraction and Ramanujan's continued fraction of order four

Shruthi C. Bhat, B. R. Srivatsa Kumar

Abstract

By employing the classical tools from the theory of $q$-series and theta functions, new fascinating identities on different continued fractions can be achieved. In this article, we use the product expansion of Jacobi's theta function to establish identities that connect Ramanujan-Göllnitz-Gordon continued fraction with Ramanujan's continued fraction of order four. Also, we obtain Eisenstein series identities using Ramanujan's $_1 ψ_1$ summation formula.

On Eisenstein series identities and new identities connecting Ramanujan-Göllnitz-Gordon continued fraction and Ramanujan's continued fraction of order four

Abstract

By employing the classical tools from the theory of -series and theta functions, new fascinating identities on different continued fractions can be achieved. In this article, we use the product expansion of Jacobi's theta function to establish identities that connect Ramanujan-Göllnitz-Gordon continued fraction with Ramanujan's continued fraction of order four. Also, we obtain Eisenstein series identities using Ramanujan's summation formula.

Paper Structure

This paper contains 5 sections, 6 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Definitions and Preliminary results
  3. Identities connecting Ramanujan-Göllnitz-Gordon continued fraction and Ramanujan's continued fraction of order four
  4. Eisenstein series identities using Ramanujan's $_1\psi_1$ summation formula
  5. Conclusion

Key Result

Lemma 2.1

We have and

Theorems & Definitions (12)

  • Lemma 2.1: Adiga_1985
  • Theorem 3.1
  • proof : Proof of \ref{['O1-O3']}
  • proof : Proof of \ref{['33O3-11O1']}
  • proof : Proof of \ref{['3O3-1O1']}
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • Theorem 4.3
  • proof
  • ...and 2 more