The Rogers-Ramanujan dissection of a theta function

TL;DR

This paper generalizes Ramanujan's identities involving theta functions and Rogers-Ramanujan functions, revealing new dissections and relations that extend beyond classical modular forms, supported by asymptotic analysis.

Contribution

It introduces a parameterized family of identities that dissect theta functions into sums of generalized Rogers-Ramanujan functions, extending classical results and exploring their modular properties.

Findings

Derived an infinite family of identities generalizing Ramanujan's relation.

Showed that for s > 2, identities transcend modular forms and are more complex.

Provided asymptotic evidence explaining Ramanujan's derivation of modular relations.

Abstract

Page 27 of Ramanujan's Lost Notebook contains a beautiful identity which not only gives, as a special case, a famous modular relation between the Rogers-Ramanujan functions $G(q)$ and $H(q)$ but also a relation between two fifth order mock theta functions and $G(q)$ and $H(q)$. We generalize Ramanujan's relation with the help of a parameter $s$ to get an infinite family of such identities. Our result shows that a theta function can always be ``dissected'' as a finite sum of products of generalized Rogers-Ramanujan functions. Several well-known results are shown to be consequences of our theorem, for example, a generalization of the Jacobi triple product identity and Andrews' relation between two of his generalized third order mock theta functions. We give enough evidence, through asymptotic analysis as well as by other means, to show that the identities we get from our main result for $s>2$ transcend the modular world and hence look difficult to be written in the form of a modular relation. Using asymptotic analysis, we also offer a clinching evidence that explains how Ramanujan may have arrived at his generalized modular relation.