17 problems

Let $n,k\in\mathbb{N}$ and $m\in\mathbb{N}_0$. Laguerre polynomial divisibility conjecture. If the Laguerre polynomial $L_k$ divides the generalized Laguerre polynomial $L_n^{(m)}$…

Generalized Laguerre Galois-group conjecture. The polynomial $L_n^{(-1-n-r)}(x)$ is irreducible over $\mathbb{Q}$, and its Galois group contains the alternating group $A_n$. For a…

Let $(\alpha_n)$ be a sequence of non-integers such that … exists, with $0<A<1$. Consider the scaled Laguerre polynomials $L_n^{(\alpha_n)}(nz)$. Martínez et al.'s conjecture. The…

Resultant stability conjecture. For any positive integers $n,m$ such that $n\leq m$, and any $a>0$, the resultant $\operatorname{res}_z(P_n(s,z),P_m(s+a,z))$ is stable.

For $n=0,1,\dots$ and $\alpha>-1$, define the transitional Laguerre polynomials $\widehat{L}^{(\alpha)}_{n+1}(\cdot,t)$ by … where … Let the extreme zeros denote the smallest and l…

For $n=0,1,\dots$ and $\alpha>-1$, define the transitional Laguerre polynomials $\widehat{L}^{(\alpha)}_{n+1}(\cdot,t)$ by … where … Let the extreme zeros mean the smallest and lar…

Let $V_n^m(w_\alpha):C_u\to C_u$ be the truncated de la Vallée Poussin polynomial operator associated with the Laguerre weight $w_\alpha$, and let $\alpha$ and $\gamma$ be the para…

Convergence conjecture. For all $alpha-1$,

Let $n\ge1$, $\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb N^n$, and $a=(a_1,\dots,a_n)\in(0,+\infty)^n$ be as in the Laguerre-integral conjecture, and let $K_\alpha(a)$ denote the c…

Laguerre-integral lower-bound conjecture. Assuming $0<a_{1}\le\dots\le a_{n}$, one has

Let $\alpha>-1$, let $\lambda$ be an even partition, and let $\Omega_{\lambda,\mu}^{(\alpha)}$ be the polynomial whose zeros determine the exceptional zeros of…

For integers $r\geq 0$ and $n\geq 1$, define … Let $\mathcal{L}_{n}^{\langle r \rangle}(x):=n!L_{n}^{\langle r \rangle}(x)$, and let $G_n(r)$ denote the Galois group of…

For nonnegative integers $a,b,c,d$ satisfying $a+b=c+d=S$, let $Q(a,b,c,d)$ denote the corresponding entry of the matrix $\mathcal{Q}_S$. Middle-row minimum conjecture. One has … E…

Let $b1>-1$, let $bb$ be an even partition, and let $bc$ be the partition parameter occurring in the generalized Laguerre polynomial $a9_{bb,bc}^{(b1)}$. The zeros of…

Let $L^{I,\,\delta}_{1,n'}(y)$ be the type I exceptional Laguerre polynomial, with $\delta=d-1/2$ and degree $N=n'+1$, obtained in the construction above. The parameters $n'$ and…

Let $Q(x)$ be a weight function defining the Laguerre-type orthogonal polynomials considered in the paper. The contours referred to are the contours used in the asymptotic construc…

Let $g^{(\alpha)}_{\mu}(z)$ be the function defined by the Wronskian construction of the rationally extended radial oscillator potential, with $\mu$ as specified by the multi-index…