A new construction of Ewell's octuple product identity (Q1773227)

The authors construct an octuple identity in the form \[ \begin{aligned}\prod_{n=1}^\infty &(1-q^n)^2(1-aq^n)(1-a^{-1}q^{n})(1-aq^{n-1})(1-a^{-1}q^{n-1}) \\ &\times (1-a^2q^{2n-1})(1-a^{-2}q^{2n-1})\\ &=2P(q) \sum_{n=-\infty}^\infty (-1)^na^{4n}q^{2n^2}-Q(q)\sum_{n=-\infty}^\infty a^{2n+1}(-q)^{n(n+1)/2},\tag{1}\end{aligned} \] where \[ P(q) =\prod_{n=1}^\infty (1-q^{4n})\quad {\text{and}}\quad Q(q) = \prod_{n=1}^\infty (1-(-q)^n). \] They give two proofs of (1) in this article. The first proof is inspired by the proof of the quintuple product identity given by L. Carlitz and M.V. Subbarao. The identity (1) is established by first multiplying both sides of (1) by \(\prod_{n=1}^\infty (1-q^{2n})\). The authors then apply the triple product identity three times and derive (1) by manipulating the resulting series. For the second proof. the authors expand the left hand side of (1) as a power series in the form \[ \sum_{n=-\infty}^\infty A_n(x)a^n. \] Using some properties of the product on the left hand side of (1), they determine certain recurrences satisfied by \(A_n(x)\) and determine the explicit form of \(A_n(x)\).