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

TL;DR

This paper extends the Ramanujan-Mordell formula using modular forms to include coefficients 1 and p, providing new identities for all k > 1 and 0 ≤ j ≤ k involving primes p.

Contribution

It introduces a novel extension of the Ramanujan-Mordell formula leveraging modular form properties and Fourier expansions at all cusps of b3_0(4p).

Findings

Derived a new formula involving modular forms and primes p.

Computed Fourier series expansions at all cusps of b3_0(4p).

Extended the classical Ramanujan-Mordell formula to broader cases.

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 $ 1 < k \in \mathbb{n} $, $0 \leq j \leq k$ and $p$ an odd prime. We obtain this result by computing the Fourier series expansions of modular forms at all cusps of $\Gamma_0(4p)$.