IJPAM: Volume 74, No. 4 (2012)

NEW MODULAR RELATIONS FOR
RAMANUJAN'S PARAMETER $\mu(q)$

M.S. Mahadeva Naika$^1$, B.N. Dharmendra$^2$, S. Chandankumar$^3$
$^{1,2,3}$Department of Mathematics
Bangalore University
Central College Campus
Bangalore, 560 001, INDIA


Abstract. In his `lost' notebook, S. Ramanujan introduced the parameter $\mu(q):=R(q)R(q^4)$ related to the Rogers - Ramanujan continued fraction $R(q)$. In this paper, we establish some new $P-Q$ modular equations of degree 5. We establish some general formulas for the explicit evaluations of the ratios of Ramanujan's theta function $\varphi$. We obtain several new modular relations connecting $\mu(q)$ with $\mu(q^n)$ for different positive integer $n>1,$ reciprocity theorems and also compute several new explicit evaluations.

Received: August 18, 2011

AMS Subject Classification: 33D20, 11F55, 11S23

Key Words and Phrases: continued fraction, modular equation, reciprocity theorem, theta function

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 74
Issue: 4