Relations between higher level Hurwitz class numbers

We connect generalizations of the classical Hurwitz class numbers coming from two different frameworks: one introduced by Pei and Wang, arising from the generalized Cohen–Eisenstein series, and another by Li, Skoruppa, and Zhou, arising from Eichler orders of quaternion algebras. As applications, we obtain new basis for Eisenstein space E 3 / 2 + ( 4 N , id ) , a generalization of recent results of Beckwith and Mono, and a generalization of Gauss' formula.