Generalizing Ramanujan's J Functions
Number Theory
2011-10-04 v2
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.
Keywords
Cite
@article{arxiv.1109.5957,
title = {Generalizing Ramanujan's J Functions},
author = {Jerome Malenfant},
journal= {arXiv preprint arXiv:1109.5957},
year = {2011}
}
Comments
9 pages; minor changes in wording and in a section heading