Guided by an approach used in the study of membranes we construct the partition function for a discrete Gaussian chain model of polymers. The Hamiltonian is , where describes the chain connectivity, is the inertia tensor, which gives the shape and size of the polymer chain, and tr denotes the trace. Here we consider only the size, , which is obtained from the partition function by differentiating with respect to the conjugate variable . The partition function is expressed in terms of the hypergeometric function , an orthogonal polynomial, where n is the number of monomers and is the ratio of and the coefficient in . The limit corresponds to the random coil, while the limit describes a compact object. We also study the excluded volume problem by discretizing Edwards' continuous self-avoidance term. We obtain dimensionally regularized expressions for the radius of gyration and the end-to-end distance. The short distance cut-off dependence of the continuous model is reproduced.