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 Delta and Delta p(2), for p a prime. Employing this identity, we extend the results of Toh [8] 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 [4] 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. (C) 2015 Elsevier Inc. All rights reserved.