Modular Equations and Theta—Function Identities in Notebook 1

Abstract

Chapters 19-21 in Ramanujan’s second notebook are devoted almost exclusively to modular equations (Part III [3, pp. 220–488]). Ramanujan clearly loved modular equations, and, as the content of Chapters 34 and 35 makes manifest, he found many applications of these equations. Thus, it is surprising that the first notebook contains several dozen modular equations that he failed to record in his second notebook. Some of these are easy to prove with the help of modular equations in the second notebook, and so Ramanujan might have considered them less important and not worthy of repeating in his second notebook. However, many of them are apparently not so easy to prove. Some have degrees not examined in the second notebook. For example, on page 298 Ramanujan records a modular equation of degree 49. Not only does Ramanujan not consider modular equations of this degree in his second notebook, but apparently no one else had found a modular equation of degree 49 up until that time. Even at this writing, we know of no other modular equation of degree 49.