On a generalized identity connecting theta series associated with discriminants $Δ$ and $Δp^2$

When the discriminants $Δ$ and $Δp^2$ 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 $Δ$ and $Δp^2$ 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 $Δ$ to a theta series associated to a subset of binary quadratic forms of discriminant $Δp^2$. Here and everywhere $p$ is a prime.

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