31 problems

Somos's conjecture. The sequence $\{\Delta_n\}_{n\in\mathbb{N}}$ satisfies half of the Somos-4 recurrence

Let $\{s_n\}_{n \in \mathbb{N}}$ be a sequence satisfying a particular convolution recursion, and let $\{H_n\}_{n \in \mathbb{N}}$ be its sequence of Hankel determinants. Barry's c…

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

Let $d\in\mathbb{N}_{\geq 2}$, let $\zeta$ be a primitive $d$th root of unity, and let $A_d$ be the set defined earlier in the paper. Let $\mathcal H(n,2\zeta)$ denote the correspo…

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…

Chapoton–Han's determinant conjecture. The determinant satisfies

Conjecture 7.6. The sixth Hankel determinant satisfies

Let $f$ be a univalent function with inverse expansion … For an analytic function $g(z)=z+\sum_{n=2}^{\infty}a_nz^n$, write … Let $\mathcal{R}$ denote the class of bounded turning…

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

Fu's periodicity conjecture. The sequence of Hankel determinants

Conjecture for $\tilde{c}_1(\alpha)$. Comparing the relevant asymptotic formulas suggests that

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

Dougherty–French–Saderholm–Qian conjecture. The sequence $(H_n)_{n\ge 0}$ satisfies a linear recurrence with constant coefficients of order $2^d$.

Let $f\in\mathscr{S}^*_{\wp}$, where $\mathscr{S}^*_{\wp}$ is the class of normalized analytic functions satisfying … and let $H_3(1)$ denote the third Hankel determinant of the Ta…

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 $f_p(z)=\prod_{i=0}^{\infty}P(z^{p^i})$, where $p\geq 2$, $P(z)=v_0+v_1z+\cdots+v_{p-1}z^{p-1}$, $v_0=1$, and $v_i\in\{-1,1\}$ for $i=1,\ldots,p-1$. Let $N_p^{\prime}$ be the n…

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 $(\mu(n))_{n\ge 0}$ be a sequence with $\mu(0)=1$, and assume that the Hankel determinants … are nonzero. Define polynomials … by the determinant formula given in the source. F…