Extensions of Ramanujan-Mordell formula with coefficients $1$ and $p$
Number Theory
2018-08-06 v1
Abstract
We use properties of modular forms to prove the following extension of the Ramanujan-Mordell formula, \begin{align*} z^{k-j}z_p^{j}=&\frac{p_{\chi}^{k-j}-1}{p_{\chi}^{k}-1}F_p(k,j;\tau)+ \frac{p_{\chi}^{k}-p_{\chi}^{k-j}}{p_{\chi}^{k}-1}F_p(k,j;p\tau)+z^{k} A_p(k,j;\tau), \end{align*} for all , and an odd prime. We obtain this result by computing the Fourier series expansions of modular forms at all cusps of .
Cite
@article{arxiv.1808.01049,
title = {Extensions of Ramanujan-Mordell formula with coefficients $1$ and $p$},
author = {Zafer Selcuk Aygin},
journal= {arXiv preprint arXiv:1808.01049},
year = {2018}
}