ANNALI DELL'UNIVERSITA' DI FERRARA

, Volume 64, Issue 1, pp 165–183 | Cite as

Some results on a special case of a general continued fraction of Ramanujan

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Abstract

We derive a new special case C(q) of a general continued fraction recorded by Ramanujan in his Lost Notebook. We give a representation of the continued fraction C(q) as a quotient of Dedekind eta-function and then use it to prove modular identities connecting C(q) with each of the continued fractions \(C(-q)\), \(C(q^{2})\), \(C(q^{3})\), \(C(q^{5})\), \(C(q^{7})\), \(C(q^{11})\), \(C(q^{13})\) and \(C(q^{17})\). We also prove general theorems for the explicit evaluation of the continued fraction C(q) by using Ramanujan’s class invariants.

Keywords

Continued fraction Modular equations Explicit values Ramanujan’s class invariants 

Mathematics Subject Classification

33D15 11A55 30B70 

Notes

Acknowledgements

The first author (N. Saikia) is thankful to Council of Scientific and Industrial Research of India for partially supporting the research work under the Research Scheme No. 25(0241)/15/EMR-II [F. No. 25(5498)/15]. The authors also thank anonymous refree for his/her valuable suggestions and comments.

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Copyright information

© Università degli Studi di Ferrara 2017

Authors and Affiliations

  1. 1.Department of MathematicsRajiv Gandhi UniversityDoimukhIndia

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