Some results on Ramanujan’s continued fractions of order ten and applications

Shraddha Rajkhowa () and Nipen Saikia ()
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Shraddha Rajkhowa: Rajiv Gandhi University
Nipen Saikia: Rajiv Gandhi University

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 1, 13-37

Abstract: Abstract We derive two continued fractions I(q) and J(q) of order ten from a general continued fraction identity recorded by Ramanujan in his notebook. We prove theta-function identities for the continued fractions I(q) and J(q). Using Ramanujan’s parameter $$k(q)=R(q)R^2(q^2)$$ k ( q ) = R ( q ) R 2 ( q 2 ) , where R(q) is the Rogers-Ramanujan continued fraction, together with a new parameter u(q), we prove general theorems for the explicit evaluations of $$I(\pm q)$$ I ( ± q ) and $$J(\pm q)$$ J ( ± q ) and give examples. As applications of some of the identities of I(q) and J(q), we derive some partition identities using colour partition of integers. We establish 2-dissections for the continued fraction $$I^*(q):=q^{-3/4}I(q)$$ I ∗ ( q ) : = q - 3 / 4 I ( q ) and $$J^*(q)=q^{-1/4}J(q)$$ J ∗ ( q ) = q - 1 / 4 J ( q ) and their reciprocals.

Keywords: Continued fraction; Theta-functions; Explicit values; Dissection formulas; Partition of integer; Colour partition; 30B70; 11F27; 11A55; 11P84 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00456-5

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