An identity connecting theta series associated with binary quadratic forms of discriminant \(\Delta\) and \(\Delta(\mathrm{prime})^2\) (Q2352021)

The author considers positive definite binary quadratic forms of discriminant \(\Delta\) and \(\Delta p^2\) for a prime \(p\). The main result is an identity which connects the theta series of the forms with discriminant \(\Delta\) with those of discriminant \(\Delta p^2\) if these discriminants are ``idoneal'', i.e., the genera of the forms consist only of one class. The proof handles all idoneal forms case by case. This result is used to write the theta series of forms of discriminant \(\Delta p^2\) as a linear combination of Lambert series. This Lambert series decompositions are then used to give explicit representation formulas for forms of discriminants \(\Delta p^2\), explicitly written down for \(\Delta p^2= 36,75\) and \(180\). The main identity is generalized without proof to discriminants which are not necessarily idoneal. A proof is said to appear in a subsequent paper.