On the expansion of a continued fraction of Gordon (Q1597191)

The infinite product \[ P=\frac{(q^3;q^8)_\infty (q^5;q^8)_\infty}{(q;q^8)_\infty (q^7;q^8)_\infty} \] has a very nice and simple continued fraction expansion. In this paper Hirschhorn proves that the power series expansions of both \(P\) and \(1/P\) have beautiful regularities. Indeed, if \(P=\sum a_nq^n\) and \(P^{-1}=\sum b_nq^n\), then \[ a_{8n}>0,\quad a_{8n+1}>0,\quad a_{8n+2}>0,\quad a_{8n+3}=0, \] \[ a_{8n+12}>0,\quad a_{8n+5}<0,\quad a_{8n+6}<0,\quad a_{8n+7}=0, \] \[ b_{8n}>0,\quad b_{8n+1}<0,\quad b_{8n+2}=0,\quad b_{8n+3}>0, \] \[ b_{8n+4}<0,\quad b_{8n+5}>0,\quad b_{8n+6}=0,\quad b_{8n+7}<0 . \]