An identity connecting theta series associated with binary quadratic forms of discriminant Δ and [formula omitted

Abstract

We state and prove an identity which connects theta series associated with binary quadratic forms of idoneal discriminants Δ and Δp2, for p a prime. Employing this identity, we extend the results of Toh [8] by writing the theta series of forms of discriminant Δp2 as a linear combination of Lambert series. We then use these Lambert series decompositions to give explicit representation formulas for the forms of discriminant Δp2. Lastly, we give a generalization of our main identity, which employs a map of Buell [4] to connect forms of discriminant Δ to Δp2. Our generalized identity links theta series associated with a single form of discriminant Δ to a theta series associated with forms of discriminant Δp2, where Δ and Δp2 are no longer required to be idoneal.