14 problems

Let $\overline{h}_k(x)=\nu_k\,p_{k,1}(x)\,p_{k,2}(x)\cdots p_{k,\alpha}(x)=\nu_k\,\widetilde{p}_k(x)$, where each $p_{k,n}$ is monic and $\nu_k$ is rational. Write the OEIS sequenc…

Let $A_{\mathcal K,k,m}(0)$ denote the constant term for the second family of meromorphic modular forms. K-family prime-parameter conjecture. The following assertions hold: if…

Let $\mathcal{K}$ and $\overline{\mathcal{K}}$ be the two families of meromorphic modular forms in the source, and let $A_{\mathcal{K},k,m}(0)$ and…

Double-heptagon connection-point conjecture. The point $P$ is a connection point if and only if $N$ is even and $[\overline{x}:\overline{y}]$ belongs to the orbit of…

Modulo-two orbit conjecture. An element $s=\frac{x}{y}\in\mathbb{Q}[\lambda_7]$ belongs to $H_7\cdot\infty$ if and only if $[\overline{x}:\overline{y}]$ belongs to the orbit of…

Brent's cusp-coefficient nonvanishing conjecture. There are corresponding polynomials $T_n\in\mathbb{Q}[x]$ satisfying $\tau_m(n)=T_n(m)$ for $m=3,4,\ldots$, together with the fact…

Let $H_{6,m}$ have Fourier expansion … where $q_m(z)=\exp(2\pi i z/\lambda_m)$ and $m\geq3$. Brent's $H_{6,m}$ interpolation conjecture. For every $n$, there is a polynomial…

Let $H_{4,m}$ have Fourier expansion … where $q_m(z)=\exp(2\pi i z/\lambda_m)$ and $m\geq3$. Brent's $H_{4,m}$ interpolation conjecture. For every $n$, there is a polynomial…

Let $a_n(x)$ be the irreducible factor occurring in the conjectured factorization of $A_n(x)$. Brent's root-location conjecture. If $n>1$, $a_n(\rho)=0$, and $\rho\neq\pm2$, then ……

Brent's nonvanishing conjecture. For each fixed integer $n\geq -1$ and every integer $m\geq 3$, one has

Brent's interpolation conjecture. For every integer $n>-2$, there exists a polynomial $A_n(x)\in\mathbb{Q}[x]$ such that $\alpha_n(m)=A_n(m)$ for $m=3,4,\ldots$, with…

Let $k\geq 5$, let $W$ be the relevant word set, let $\nu^{(k)}$ be the coefficient function, let $L_p\in\Lambda_p$ be the linear recurrence associated with the coefficients, and l…

Let $p=p_\lambda$ be the polynomial family considered in the paper, and let $\gamma(p)$ denote the quantity defined in the preceding observation. Parity formula for $p_\lambda$. Fo…

Let $G_k$ be the Hecke group with parameter $k$, let $W$ denote the words under consideration, and let $\nu_1,\ldots,\nu_8$ be the functions defined in Observation 1, using the Fib…