Extensions of Ramanujan-Mordell formula with coefficients $1$ and $p$

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_χ^{k-j}-1}{p_χ^{k}-1}F_p(k,j;τ)+ \frac{p_χ^{k}-p_χ^{k-j}}{p_χ^{k}-1}F_p(k,j;pτ)+z^{k} A_p(k,j;τ), \end{align*} for all $1<k\in n$, $0\le j\le k$ and $p$ an odd prime. We obtain this result by computing the Fourier series expansions of modular forms at all cusps of ${\Gamma }_0(4p)$.