19 problems

Let $C(x)=\sum_{n\geq 0}C_nx^n$ be the Catalan generating function, let $F(x,r)=C(x)^r$, and write $H_n(F(x,r))$ for the $n\times n$ Hankel determinant of the coefficients of…

Hankel determinant conjecture. For every integer $n\ge3$,

Forrester–Frankel conjecture. As $N\to\infty$ with $m$, $q$, and $\mu$ fixed,

Let $(n_k)$ be the sequence defined earlier in the paper, and let $\mathcal H(n,1)$ be the associated Hankel determinant. Monotonicity conjecture. The sequence of absolute values…

Let $\mathcal H(n,1)$ denote the Hankel determinant studied in the paper. For each fixed integer $k\geq 0$, consider the values with index $n=2^m+k$. Asymptotic product formula con…

Conjecture 7.6. The sixth Hankel determinant satisfies

Let $F(x,r)=M(x)^r$, where $M(x)$ is the generating function of the Motzkin numbers, and let $H_n(F(x,r))$ be its Hankel determinants. The conjecture asserts that, for…

Let $M(x)=\sum_{n\geq 0}M_nx^n$ be the generating function of the Motzkin numbers, let $F(x,r)=M(x)^r$, and write $H_n(F(x,r))$ for the Hankel determinant of the coefficients of…

Let $M(x)=\sum_{n\geq 0}M_nx^n$ be the generating function of the Motzkin numbers, let $F(x,r)=M(x)^r$, and write $H_n(F(x,r))$ for the Hankel determinant of the coefficients of…

The case $\ell\geq n+2$. The sequence $\Delta_j^{(n+2)}$ is $2n(n+1)$-periodic when $n$ is even and $2n(n+1)$-antiperiodic, hence $4n(n+1)$-periodic, when $n$ is odd, and its terms…

Hankel determinant periodicity and recurrence conjecture. The $k+2$ sequences $\Delta_n^{(0)},\Delta_n^{(1)},\dots,\Delta_n^{(k+1)}$ take only the values $-1,0,1$. They are…

Kumar–Gangania conjecture. If $f\in\mathcal{S}_{\wp}^{*}$, then the sharp bound is

Extremal-set periodicity conjecture. The sequence $H(D^{(m,V)})$ is periodic and has the form

Sharp-value conjecture. If $f\in\mathcal F$, then

Let $L$ be the linear functional in the preceding generalization, with integer parameters $\alpha$, $\beta_i$, and $\gamma_i$, and let $P_n/Q_n$ be the associated rational approximant…

Let $\Delta_n$ be the Hankel determinants whose moments are defined by the recurrence in the source. Let $\tilde{\mathcal C}$ be the quintic curve … which is isomorphic to the rele…

Let $G(x,r)$ be the generalized generating function used in the paper, and write $H_n(G(x,r))$ for the Hankel determinant of its coefficients. For a positive integer $t$, consider…

Let $C(x)=\sum_{n\geq 0}C_nx^n$ be the Catalan generating function, let $F(x,r)=C(x)^r$, and write $H_n(F(x,r))$ for the $n\times n$ Hankel determinant of the coefficients of…

Conjectured properties of the Hankel determinants. The following properties hold: