P–Q “mixed” modular equations of degree 15

Abstract

Ramanujan in his second notebook recorded total of seven P–Q modular equations involving theta–function \(f(-q)\) with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving his theta–functions \(\varphi (q)\) and \(\psi (-q)\) with moduli of orders 1, 3, 5 and 15. As a consequence, several values of quotients of theta–function and a continued fraction of order 12 are explicitly evaluated.