We generalize the construction from (Roehrig and Zwegers in Int J Number Theory, 18(7):1491–1515, 2022) of theta series for quadratic forms of signature \((n-1,1)\) with homogeneous and spherical polynomials. Namely, we allow that the parameters \({\textbf {c}}_1,{\textbf {c}}_2\) , which define the theta series and ensure the convergence of the defining series, are located on the boundary of the cone \(C_Q\) . This enables us to study several interesting examples such as Eisenstein series, modular forms on \(\Gamma _0(4)\) which appear during the investigation of quadratic polynomials of a fixed discriminant, and a mock theta function of order 2 that is connected to the generating function of the Hurwitz class numbers \(H(8n+7)\) .
