Generalized Glasby–Paseman sequence conjecture

For positive integers $m$ and $l$, and a positive real number $a$, define the sequence

$$\left\{\frac{\sum_{i=0}^{r}\{\binom{m}{i}a^{i}\}^{l}}{\sum_{i=0}^{r}\{\binom{r}{i}a^{i}\}^{l}}\right\}_{0\leq r\leq m}.$$

Generalized Glasby–Paseman sequence conjecture. This sequence satisfies the following properties:

1. It is unimodal for all $m,l\in\mathbb{Z}^{+}$ and $a\in\mathbb{R}^{+}$.
2. For $l\in\mathbb{Z}^{+}$ and $a\in\mathbb{R}^{+}$, it is log-concave for all but finitely many $m\in\mathbb{Z}^{+}$.
3. For $l,a\in\mathbb{Z}^{+}$ and $m\in\mathbb{Z}_{\geq 2}$, it attains its unique maximum at an index $r$ equal to one of

$$\left\lfloor\frac{am+a+2}{2a+1}\right\rfloor-1,\qquad \left\lfloor\frac{am+a+2}{2a+1}\right\rfloor,\qquad \left\lfloor\frac{am+a+2}{2a+1}\right\rfloor+1.$$

4. For $l\in\mathbb{Z}^{+}$ and $a\in\mathbb{R}^{+}$, its maximum value is asymptotic to

$$\frac{l^{\frac{1}{2}}}{\sqrt{2\pi m}}\cdot\frac{(1+2a)^{\frac{1}{2}}(1+a)a^{\frac{l-2}{2}}}{(1+a)^l-1}\cdot\left(\frac{1+2a}{1+a}\right)^{(m+\frac{1}{2})l}.$$

Here, “asymptotic” means that the ratio between the maximum value and the displayed expression approaches $1$ as $m$ approaches infinity.

These assertions extend questions about unimodality, log-concavity, the location of the maximum, and asymptotic maximum values from the Glasby–Paseman sequence to this two-parameter family. They are based on computer experiments, and the supplied text gives no resolution, so they remain open.