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 polygonal numbers with a prescribed number of repeats of each distinct polygonal number. We compare such representations with classical polygonal numbers, and those representations with generalized polygonal numbers. Our main result is that representations with classical polygonal numbers are equidistributed in the sense that the number of representations in the nonnegative quadrant in four-dimensional space is asymptotically \(\frac{1}{16}\) $\frac{1}{16}$ of the representations in the entire space.