368 problems

Ismail's conjecture. The sequence $(P_n)_{n\geq 0}$ satisfies

Let $\{P_n(x)\}_{n\geq 0}$ be monic polynomials orthogonal on $\mathbb{R}$ with respect to the weight … where $\rho>-1$ and $m>0$. Write $\beta_n$ for the recurrence coefficients i…

The two dual hypergeometric random walks are recurrent when $0<\delta\leq 1$, and there are no mass points in the corresponding measure. Null recurrence conjecture. This recurrence…

Root-distribution conjecture. For each $k\geq -1$, the polynomial $P^{(\lambda)}_{l+4+k}$ is square free. Furthermore, for sufficiently large $k$, it has a fixed number of roots ou…

Ignjatović's conjecture. If

Let $\mu$ be a probability measure on $\mathbb{T}$ whose density satisfies $d\mu'\geq\delta$ almost everywhere, and write … Let $\phi_n(z;d\mu)$ denote the orthonormal polynomials…

Let $f$ be a polynomial with no zeros in the closed bidisk $\overline{\mathbb{D}^2}$, and let its orthogonal polynomial approximants (OPAs) be formed in $H^2(\mathbb{D}^2)$. Weak S…

Let $f$ be a polynomial, and let its orthogonal polynomial approximants (OPAs) to $1/f$ in $H^2(\mathbb{D}^2)$ be given. Shanks's conjecture. For every polynomial $f$, the…

Let $w$ be the density of the absolutely continuous part of a measure on the unit circle, let $\alpha=(\alpha_n)_{n\geq 0}$ be its Verblunsky-coefficient sequence, and let $S$ be t…

Let $w_M$ be a weight supported on $(-1,1)$ such that $w_M(x)/(1+x)^{\beta}$ has a positive finite limit as $x\to -1$, and $w_M(x)/[-(1-x)^{\alpha}\log(1-x)]$ has a positive finite…

Let $P_n^{(\alpha,\beta)}(x)$ denote the Jacobi polynomial of degree $n$, with $n\in\mathbb{N}$, $\alpha>-1$, and $\beta>-1$. Askey's conjecture. The zeros of…

Macdonald–Stanley conjecture. The expansion coefficients $c_{\lambda,\mu}(\alpha)$ are polynomials in the parameter $\alpha$ with integer coefficients.

Let $\alpha$ be a nonnegative integer, and let $\{L_n^{\alpha,M,N}(x)\}_{n=0}^{\infty}$ be the generalized Laguerre polynomials with parameters $M$ and $N$. Differential-equation c…

For $k\geq0$ and $\alpha,\beta\geq-\frac12$, let … In the parameter range $\alpha\geq\beta$, the source had obtained an upper bound of order $O(\alpha^{2/3}(1+\alpha/k)^{1/3})$. Kr…

Let $A,A^*$ be a tridiagonal pair on a finite-dimensional vector space $V$ over an algebraically closed field $\mathbb K$. Let $d$ be its diameter, and let…

Let $\mu$ be a probability measure on the unit circle with infinite support, decomposed as … and let $\alpha=(\alpha_n)$ be its Verblunsky coefficients. For distinct angles…

Let $\mu\in C[-1,1]$ be a nonnegative majorant, and let $\omega_\mu\in\mathcal{P}_n$ be the snake polynomial associated with $\mu$, meaning that $|\omega_\mu(x)|\leq\mu(x)$ on…

Ismail's conjecture. If

Let $P_n(z)$ be the orthonormal polynomials associated with the Jacobi operator, and let the asymptotic formulas for $P_n(z)$ have coefficients as in the complex spectral asymptoti…

Let $\\{\alpha_n\\}_{n\in\mathbb{Z}_+}$ be Verblunsky coefficients satisfying … for some $\gamma\in(0,1)$, and let the associated Szegő matrices be evaluated at…

Nonvanishing normal derivative conjecture. For every integer $t\geq 0$,

Let a sequence of orthogonal polynomials be defined using a weight function on a contour connecting the endpoints of the orthogonality interval, and suppose that the weight functio…

Let $P_{\mathcal{D},n}(\eta;\boldsymbol{\lambda})$ be the multi-indexed Meixner–Pollaczek or continuous Hahn polynomial, with parameters $\boldsymbol{\lambda}$ and index set…

Let $p_n$ be the polynomials defined by formula (20) or, equivalently, by the recurrent relation (10) in the source. Reality conjecture. All zeros of $p_n$ are real. The distributi…

Minimal-order recurrence conjecture. The minimal degree of an element of $\mathcal{S}$, and hence the minimal order of the recurrence relation for the exceptional Hermite polynomia…