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

TL;DR

This paper generalizes Patane's theorem by establishing a new identity linking theta series of binary quadratic forms for discriminants elta and elta p^2 without the idoneal restriction, involving subsets of forms.

Contribution

It extends Patane's result to non-idoneal discriminants, connecting theta series of quadratic forms of discriminants elta and elta p^2 through a novel identity.

Findings

Established a new identity linking theta series for elta and elta p^2

Removed the idoneal restriction in the theorem

Connected theta series of forms of discriminant elta to subsets of forms of discriminant elta p^2

Abstract

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