On representation of an integer as the sum of three squares and ternary quadratic forms with the discriminants p2, 16p2

Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p2n)−ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants p2, 16p2. These forms are related by Watsonʼs transformations. To prove this identity we employ the Siegel–Weil and the Smith–Minkowski product formulas.