## On a generalized identity connecting theta series associated with discriminants $\varDelta $ and $\varDelta p^2$

### Volume 176 / 2016

Acta Arithmetica 176 (2016), 343-364
MSC: 11E16, 11E25, 11F27, 11H55, 11R29.
DOI: 10.4064/aa8383-6-2016
Published online: 28 October 2016

#### Abstract

When the discriminants $\varDelta $ and $\varDelta p^2$ have one form per genus, Patane (2015) proves a theorem which connects the theta series associated to binary quadratic forms of each discriminant. This paper generalizes the main theorem of Patane (2015) by allowing $\varDelta $ and $\varDelta p^2$ to have multiple forms per genus. In particular, we state and prove an identity which connects the theta series associated to a single binary quadratic form of discriminant $\varDelta $ to a theta series associated to a subset of binary quadratic forms of discriminant $\varDelta p^2$. Here and everywhere $p$ is a prime.