Results of Hurwitz type for five or more squares (Q1850939)

In this article, many interesting identities are recorded for the first time. These identities are related to the sums of squares function, and they are motivated by Ramanujan's identity \[ \sum_{n\geq 0} p(5n+4)q^n = 5\prod_{n\geq 1}\frac{(1-q^{5n})^5}{(1-q^n)^6}. \] One of the simplest examples is \[ \sum_{n\geq 0} r_7(24n+23)q^n=49728\prod_{n\geq 1}\frac{(1-q^{2n})^{10}(1-q^{3n})^3}{(1-q^n)^6}, \] where \(r_k(n)\) denotes the number of representations of \(n\) as a sum of \(k\) squares. Most of the identities in this paper are proved in an elementary way (but by no means obvious) using mathematical induction.