The Parents of Jacobi's Four Squares Theorem Are Unique

Jacobi’s four squares theorem asserts that the number of representations of a positive integer n as a sum of four squares is 8 times the sum of the positive divisors of n, which are not multiples of 4. A formula expressing an infinite product as an infinite sum is called a product-to-sum identity. The product-to-sum identities in a single complex variable q from which Jacobi’s four squares formula can be deduced by equating coefficients of qn (the “parents”) are explored using some amazing identities of Ramanujan, and are shown to be unique in a certain sense, thereby justifying the title of this article. The same is done for Legendre’s four triangular numbers theorem. Finally, a general uniqueness result is proved.