The conjectural recurrence and functional equation for cusp Hilbert series

For each $d\geq 0$, let $\mathit{NH}_d(t)$ be the normalized cusp Hilbert series, let ${d\brack r}_q$ be the $q$-binomial coefficient, and let $(t;q)_m$ be the finite $q$-Pochhammer symbol. The cusp Hilbert-series recurrence conjecture.

$$\mathit{NH}_d(t^2)=\sum_{r=0}^d t^r {d\brack r}_q (t;q)_{d-r}\mathit{NH}_r(tq^{d-r}),$$

and

$$q^{d^2}t^d\mathit{NH}_d(q^{-2d}t^{-1})=\mathit{NH}_d(t).$$

These identities translate the conjectural quotient formulas into statements about normalized Hilbert-series polynomials. They are supported by the computations described in the paper for small $d$, but remain conjectural in general.