A note on Shintani's invariants

Shintani’s celebrated invariants are conjectured to generate abelian extensions of real quadratic number fields, offering a potential solution to Hilbert’s 12th problem in that setting. In this note, we derive new expressions for Shintani’s invariants by generalizing an observation of Yamamoto, who showed that these invariants - originally formulated using the double sine function - can be expressed in terms of the q-Pochhammer symbol.