On a generalized identity connecting theta series associated with discriminants $\Delta$ and $\Delta p^2$

When the discriminants $\Delta$ and $\Delta p^2$ are idoneal, Patane proved a theorem which connects the theta series associated to binary quadratic forms of each discriminant. This paper generalizes the main theorem of Patane by no longer requiring $\Delta$ and $\Delta p^2$ to be idoneal. In particular, we state and prove an identity which connects the theta series associated to a single binary quadratic form of discriminant $\Delta$ to a theta series associated to a subset of binary quadratic forms of discriminant $\Delta p^2$. Here and everywhere $p$ is a prime.