Powers of Euler's product and related identities
Let \((q)_\infty := \prod_{n=1}^\infty (1-q^n)\) and \[ (q)_\infty^r = \sum_{n=0}^\infty a_r(n)q^n, r\in {\mathbb Z}. \] In 1955, \textit{M. Newman} [J. Lond. Math. Soc. 30, 488--493 (1955; Zbl 0064.28203)] proved that \[ a_r(np+r\delta) = (-p)^{(p/2)-1}a_r\left(\frac{n}{p}\right) \] when \(r=2,4,6,8,10,14\) and 26, \(p>3\) is a prime satisfying \(r(p+1)\equiv 0\pmod{24}\) and \(\delta=(p^2-1)/24\). Newman's proof of this theorem employs the theory of elliptic modular functions. In this article, the authors give elementary proofs of Newman's theorem (except for \(r=26\)) using MacDonald's identities. They then conjecture similar results for products of the form \((q)_\infty^r(q^t)_\infty^s, t=2,3,4,\) for various pairs of integers \((r,s)\). Their conjectures have recently been proved by S. Ahlgren.
scientific article; zbMATH DE number 1546947