An identity connecting theta series associated with binary quadratic forms of discriminant $\Delta$ and $\Delta($prime$)^2$

We state and prove an identity which connects theta series associated with binary quadratic forms of idoneal discriminants $\Delta$ and $\Delta p^2$, for $p$ a prime. Employing this identity, we extend the results of Toh by writing the theta series of forms of discriminant $\Delta p^2$ as a linear combination of Lambert series. We then use these Lambert series decompositions to give explicit representation formulas for the forms of discriminant $\Delta p^2$. Lastly, we give a generalization of our main identity, which employs a map of Buell to connect forms of discriminant $\Delta$ to $\Delta p^2$. Our generalized identity links theta series associated with a single form of discriminant $\Delta$ to a theta series associated with forms of discriminant $\Delta p^2$, where $\Delta$ and $\Delta p^2$ are no longer required to be idoneal.