Extensions of Ramanujan-Mordell formula with coefficients 1 and p

We use properties of modular forms to prove the following extension of the Ramanujan-Mordell formula,zk−jzpj=pχk−j−1pχk−1Fp(k,j;τ)+pχk−pχk−jpχk−1Fp(k,j;pτ)+zkAp(k,j;τ), for all 1