A $q$-analog of Jacobi's two squares formula and its applications

Abstract

We consider a $q$-analog $r_2(n,q)$ of the number of representations of an integer as a sum of two squares $r_2(n)$. This $q$-analog is generated by the expansion of a product that was studied by Kronecker and Jordan. We generalize Jacobi's two squares formula from $r_2(n)$ to $r_2(n,q)$. We characterize the signs in the coefficients of $r_2(n,q)$ using the prime factors of $n$. We use $r_2(n,q)$ to characterize the integers which are the length of the hypotenuse of a primitive Pythagorean triangle.