Representations of integers as sums of four polygonal numbers and partial theta functions

In this paper, we consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations with non-negative parameters (hence counting the number of points in a regular $m$-gon) is asymptotically the same as $\frac{1}{16}$ times the number of such representations with arbitrary integer parameters (often called generalized polygonal numbers).