The explicit normalized Hilbert-series polynomial conjecture for the cusp

Let $A$ be a commutative ring with $1$, let $q\in A$, and let $\mathit{NH}_d(t)\in A[t]$. Let ${d\brack r}_q$ be the $q$-binomial coefficient, $(t;q)_m$ the finite $q$-Pochhammer symbol, and $(t;q)_\infty$ its infinite analogue. The explicit normalized Hilbert-series conjecture. There are unique polynomials $\mathit{NH}_d(t)$ for $d\geq 0$ satisfying

$$\mathit{NH}_0(t)=1,$$ $$\mathit{NH}_d(t^2)=\sum_{r=0}^d t^r {d\brack r}_q(t;q)_{d-r}\mathit{NH}_r(tq^{d-r}),$$

and, if $\mathit{NH}_d(t)=\sum_{i=0}^d a_i t^i$,

$$a_{d-i}=q^{d(d-2i)}a_i\quad(0\leq i\leq\lfloor d/2\rfloor).$$

Moreover,

$$\mathit{NH}_d(t;q)=\sum_{j=0}^d q^{\binom{j+1}{2}+j(d-j)}c_j(q)t^j,$$

where $\sum_{j=0}^d c_j(q)t^j=(-t;q)_d$, and the coefficients are nonnegative integers. With $H_d(t;q)=\mathit{NH}_d(t;q)/(t;q)_d$, the associated series satisfies

$$\widehat Z(t;q)=\frac{1}{(tq^{-1};q^{-1})_\infty}\sum_{n=0}^\infty\frac{q^{-n^2}t^{2n}}{(q^{-1};q^{-1})_n}.$$

This packages the recurrence, functional equation, positivity, and the predicted zeta-series formula into one conjecture. It is based on computational evidence through $d\leq 30$ and is independent of the earlier recurrence conjecture.