42 problems

Let … where $a_n=(-1)^{s_2(n)}\in\{-1,1\}$ and $s_2(n)$ is the sum of the binary digits of $n$. Gawron–Miska–Ulas conjecture. For every integer $m\geq 2$, the coefficient sequence…

Let $K$ be a number field, let $d$ be a positive integer, and let … be a rational function such that each $a_i$ is the $d$-th power of an element of $K$. Pisot's d-th root conjectu…

Zudilin's conjecture. For any positive integers $N_1,N_2,\dots,N_k$,

Let … and define … Thus, $\Omega_2$ is the set of double zeros in $(0,1)$ of power series with coefficients in $\{-1,0,1\}$. Infinitely many components conjecture. The set…

Positivity conjecture. For $d\geq 2$, the trigonometric polynomials $\mathcal T_n^{(d)}(t)$ are positive for every $t\in\mathbb R$. Positivity of these polynomials is suggested as…

Let $d>2$ and define the coefficients $b_k^{(d)}$ by … For each $n\geq 0$, the corresponding partial sum is the polynomial $\sum_{k=0}^n b_k^{(d)}z^k$. Zero-free Taylor polynomial…

Let $I$ index vectors $\boldsymbol{v}_{ij}$, let $K_0$ be the monoid used to define the formal power series $\boldsymbol{z}^{\boldsymbol{k}}$, and let $\boldsymbol{v}_{ij}$ be Fano…

Non-negative coefficient conjecture. For each integer $r\geq 1$, this power series has all non-negative coefficients.

Threshold-one conjecture. The constants $\tilde\chi$ and $\tilde\chi_{\mathbb R}$ are equal to $1$.

Threshold equality conjecture. The constants $\tilde\chi$ and $\tilde\chi_{\mathbb R}$ are equal to each other.

Complementary Bell sign-run conjecture. The coefficient sequence of the power series $\exp(1-\exp(x))$ contains arbitrarily long subsequences of consecutive coefficients with the s…

Clark–Ismail's coefficient-positivity conjecture. Then $\gamma(j,k)\ge0$ for all $j>1$ and $k\ge0$. This is a coefficientwise strengthening of the positivity question for $g_j$ and…

Let $\bar{h}(\bar{x})$ be the meniscus solution expanded about $\bar{x}=0$, and let $\bar{x}_s(\theta)$ be the value given by … with $\bar{x}_s\leq0$ for $\theta\in[0,\pi/2]$ and…

For a fixed parameter $\theta$, let $\bar{h}(\bar{x})$ be the exact inverse solution of the meniscus problem, and let $\bar{x}_s(\theta)$ be the as-yet undetermined singularity clo…

Monotonicity and convexity conjecture. The sequences $b_n$ and $c_n$ are increasing and convex in $n\in\mathbb{N}_0$.

Let $u(t)=\sum_{n\geq 0}a_nt^n$ be the Majorana-transformed solution of the dimensionless Thomas–Fermi equation, and let $\tau_0$ be the location of its singularity closest to the…

Let $u(t)=\sum_{n\geq 0}a_nt^n$ be the series representing the Majorana-transformed solution of the dimensionless Thomas–Fermi equation, and let $\tau_0$ denote the singularity of…

Ziętak's coefficient-sign conjecture. It is conjectured that $c_{k,i}<0$ for every $i\geq 1$, as long as the series for $x_k$ is not reduced to a finite series.

Let $\psi(t)$ be the function defined in the preceding series representation, let $\alpha_1=1$, $\alpha_{2j}=0$, and $\alpha_{2j+1}=-\frac{2^{2j}|B_{2j}|}{(2j)!}$ for $j\in N$, and…

Radius-of-convergence conjecture. There exist integers $l_1,\ldots,l_m$ such that the radius of convergence of this series is

Let $p$ be a prime and let $r\ge0$. Define … and let … be the associated Böttcher coordinate. Asymptotic valuation and radius conjecture. For $k\equiv0\pmod p$, … Moreover, the rad…

Let … Consider the rational function $P^{-1}$ and its power-series expansion. Kauers' positivity conjecture. The power-series expansion of $P^{-1}$ has only positive coefficients.…

Let $t_m(n)$ denote the coefficient sequence associated with the power series in the paper, and let $m\in\mathbb{N}_{\geq 2}$. Unboundedness conjecture. … The preceding estimate gi…

Polynomial ramification criterion. For every positive integer $b$, there exists a polynomial $P_b\in\mathbb{Z}[x_1,\dots,x_{b+1}]$ such that $f$ is $b$-ramified if and only if…

Rationality–transcendence conjecture. The power series $F_V(x_1,\ldots,x_d)$ is either rational or transcendentally transcendental.