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

• Published in: Indian journal of pure and applied mathematics Vol. 56; no. 1; pp. 13 - 37
• Main Authors: Rajkhowa, Shraddha, Saikia, Nipen
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.03.2025
• Subjects: Mathematics; Parameters
• ISSN: 0019-5588; 0975-7465
• DOI: 10.1007/s13226-023-00456-5

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)R2(q2), 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(±q) and 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-3/4I(q) and J∗(q)=q-1/4J(q) and their reciprocals.