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We state and prove an identity which connects theta series associated withbinary quadratic forms of idoneal discriminants $\Delta$ and $\Delta p^2$, for$p$ a prime. Employing this identity, we extend the results of Toh by writingthe theta series of forms of discriminant $\Delta p^2$ as a linear combinationof Lambert series. We then use these Lambert series decompositions to giveexplicit representation formulas for the forms of discriminant $\Delta p^2$.Lastly, we give a generalization of our main identity, which employs a map ofBuell to connect forms of discriminant $\Delta$ to $\Delta p^2$. Ourgeneralized identity links theta series associated with a single form ofdiscriminant $\Delta$ to a theta series associated with forms of discriminant$\Delta p^2$, where $\Delta$ and $\Delta p^2$ are no longer required to beidoneal.

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