A Continued Fraction of Ramanujan and Some Ramanujan-Weber Class Invariants

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Adiga C., Bulkhali N. A. S., Şimşek Y., Srivastava H. M.

FILOMAT, vol.31, no.13, pp.3975-3997, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 31 Issue: 13
  • Publication Date: 2017
  • Doi Number: 10.2298/fil1713975a
  • Journal Name: FILOMAT
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.3975-3997
  • Keywords: q-Continued fractions, q-Series, Modular equations, Ramanujan-Weber class invariants, Algebraic numbers, Transcendental numbers, Dedekind eta function, EXPLICIT EVALUATIONS, NOTEBOOK
  • Akdeniz University Affiliated: Yes

Abstract

On Page 36 of his "lost" notebook, Ramanujan recorded four q-series representations of the famous Rogers-Ramanujan continued fraction. In this paper, we establish two q-series representations of Ramanujan's continued fraction found in his "lost" notebook. We also establish three equivalent integral representations and modular equations for a special case of this continued fraction. Furthermore, we derive continued-fraction representations for the Ramanujan-Weber class invariants g(n) and G(n) and establish formulas connecting g(n) and Gn. We obtain relations between our continued fraction with the Ramanujan-Gollnitz-Gordon and Ramanujan's cubic continued fractions. Finally, we find some algebraic numbers and transcendental numbers associated with a certain continued fraction A(q) which is related to Ramanujan's continued fraction F(a; b; lambda; q); the Ramanujan-Gollnitz-Gordon continued fraction H(q) and the Dedekind eta function eta(s).