Jacobi's four squares theorem asserts that the number of representations of a posi- tive 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 q n (the "par- ents") 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. 1. INTRODUCTION. A formula expressing an infinite product as an infinite series is called a product-to-sum identity. Many such identities are known. Perhaps the most famous product-to-sum identity was given by Jacobi in 1829, in his monumental work on elliptic functions, Fundamenta Nova Theoriae Functionum Ellipticarum, which is reproduced in his Gesammelte Werke (6, Vol. I, pp. 49-239); see page 234. Jacobi's formula, now known as Jacobi's triple product identity, asserts that 1 Y
['Kenneth S. Williams']
The Parents of Jacobi's Four Squares Theorem Are Unique