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

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

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