An identity connecting theta series associated with binary quadratic forms of discriminant $\Delta$ and $\Delta($prime$)^2$

TL;DR

This paper establishes a new identity linking theta series of binary quadratic forms with discriminants  and  p^2, enabling explicit formulas and generalizations beyond idoneal discriminants.

Contribution

It introduces a novel identity connecting theta series of forms with discriminants  and  p^2, extending previous results and generalizing to non-idoneal cases.

Findings

Derived explicit representation formulas for forms of discriminant  p^2.

Expressed theta series as Lambert series decompositions.

Established a generalized identity linking forms of different discriminants.

Abstract

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 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 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.