The structured discrete Hankel-transform feasibility conjecture

For the discrete Hankel-transform feasibility problem, let $(k,q)$ be $(s,\nu)$-feasible as defined in the source.

Structured discrete Hankel-transform conjecture. The following pairs are feasible:

• $\left(k,\left\lfloor\frac{\sqrt3(k^2-2k+2)}4\right\rfloor\right)$ is $(-1,0)$-feasible for every integer $k\geqslant4$;
• $\left(k,\left\lfloor\frac{k^2}4\right\rfloor\right)$ is $(-1,3)$-feasible for every integer $k\geqslant4$;
• $\left(k,\left\lfloor\frac{k^2+6k-8}8\right\rfloor\right)$ is $(-1,11)$-feasible for every integer $k\geqslant4$;
• $\left(k,\left\lfloor\frac{k^2-2}4\right\rfloor\right)$ is $(+1,5)$-feasible for every integer $k\geqslant3$.

Writing these pairs as $(k,\widetilde q_s(k,\nu))$, respectively, one has

$$k=\mathbb{A}_s^{\textup{disc}}(\widetilde q_s(k,\nu),\nu)+o(k),\qquad k\to\infty.$$

These formulas give more structured asymptotic predictions in the cases $d\in\{2,8,12,24\}$ highlighted by the source's numerical data. They are presented as conjectural refinements of the general discrete Hankel-transform conjecture.