Theta Series for Quadratic Forms of Signature $(n-1,1)$ with (Spherical)\n Polynomials

We construct almost holomorphic and holomorphic modular forms by considering\ntheta series for quadratic forms of signature $(n-1,1)$. We include homogeneous\nand spherical polynomials in the definition of the theta series (generalizing a\nconstruction of the second author) to obtain holomorphic, almost holomorphic\nand modular theta series. We give a criterion for these series to coincide,\nenabling us to construct almost holomorphic and holomorphic cusp forms on\ncongruence subgroups of the modular group. Further, we provide numerous\nexplicit examples.\n