Generalizing Ramanujan's J Functions

TL;DR

This paper extends Ramanujan's expansions of fractional-power Euler functions to prime roots greater than 3, revealing a structured pattern of non-zero functions and their relations to theta functions.

Contribution

It generalizes Ramanujan's expansions to (q^{1/N})_{ } for prime N > 3, detailing the number, form, and product of the J functions involved.

Findings

Number of non-zero J functions is (N+1)/2.

One J function has the form ±q^{X_0}.

Product of all non-zero J's is ±q^{Z}.

Abstract

We generalize Ramanujan's expansions of the fractional-power Euler functions (q^{1/5})_{\infty} = [ J_1 - q^{1/5} + q^{2/5} J_2 ](q^5)_{\infty} and (q^{1/7})_{\infty} = [ J_1 + q^{1/7} J_2 - q^{2/7} + q^{5/7} J_3 ] (q^7)_{\infty} to (q^{1/N})_{\infty}, where N is a prime number greater than 3. We show that there are exactly (N+1)/2 non-zero J functions in the expansion of (q^{1/N})_{\infty}, that one of these functions has the form +-q^{X_0}, that all others have the form +-q^{X_k} times the ratio of two Ramanujan theta functions, and that the product of all the non-zero J's is +-q^Z, where Z and the X's denote non-negative integers.