52 problems

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…

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…

Recurrence-product conjecture. The sequence satisfies

Let $p_{n+1}(z)=\sum_{i=1}^{k}Q_i(z)p_{n-i}(z)$ be a finite recurrence with fixed polynomial coefficients, and suppose $\deg p_n(z)=n$ for $n\in\mathbb N$. Let $\Xi_Q$ be the equim…

For ternary partitions, let $\operatorname{num}_{\mathcal T}(n,x)$ denote the associated numerator. Define $t(n)\coloneq\operatorname{num}_{\mathcal T}(n,1)$ for $n\geq 0$, with…

Supercritical linear-growth conjecture. For every fixed $m\ge2$, there exists a nonempty interval $1<\alpha<\alpha_m^\ast$ such that the recursion is globally well-defined and sati…

Mathar's recurrence conjecture. For every integer $n\geq 4$,

Arm-property conjecture. The identity

Formula conjecture. The solution of this recurrence is

Let a family of graphs have Laplacian matrices, and apply the procedures LaplaceExpand and SystemReduce together with the mathematical theorems described in the source. The numerat…

Let $\operatorname{Pol}_n(x)$ be the monic polynomial of degree $2n-2$ occurring in the factorization of the determinant in Theorem MS1. Polynomial-factor recurrence conjecture. Th…

Let $C^X_{r,j}$ be an entry of the $X$-circuit array, represented as a quotient of numerator and denominator polynomials. The coefficient recursions of the relevant polynomials are…

Let $C^X_{r,j}$ denote the entries of the $X$-circuit array. For $c=1,r=0$ and for each $c \ge 2$ with corresponding $r \in \{2(c-1),2(c-1)-1\}$, suppose there is a sequence of pol…

Let $C^X_r(j)$ and $C^M_r(j)$ denote the entries in rows $r$ of the corresponding circuit arrays, and let $Num$ and $Den$ denote their numerators and denominators. A sequence satis…

Triangular quilt recurrence conjecture. The recurrence

Let $(t_k)$ be the recurrence sequence counting the turns associated with the described two-firefighter strategy, with initial conditions … Recurrence containment conjecture. If th…

Minimal-order telescoping recurrence conjecture. A telescoping recurrence equation of the conjectured minimal order $m$ exists for $s\leq 10$. Computations of Perlstadt and McIntos…

Let $A^{(s)}(n)$ be the generalized Franel numbers, let $A^{(s)}_1(n)$ denote the first auxiliary sequence arising from the expansion used to obtain the Apéry limits, and define ……

Consider the original Somos-4 sequence with $x_{-1}=x_0=x_1=x_2=1$, and let $y_n^{(iv)}$ be the fourth independent shadow sequence obtained by solving the associated homogeneous eq…

Let $(d_n)$ be the unique sequence of rational numbers with initial terms … satisfying the explicit order-seven linear recurrence … where each $r_i(n)\in\mathbb{Z}[n]$ is defined b…

Conjecture. By reordering the indices if necessary, there exists an $N_0$ such that the following inequalities hold. The supplied statement span does not include the inequalities t…

Asymptotic growth conjecture. Any solution to this system grows asymptotically like

Let $P_{\mathcal{D},n}$ be the multi-indexed Meixner–Pollaczek or continuous Hahn polynomial for an arbitrary index set $\mathcal{D}$, including cases in which it is not orthogonal…

Let $X(\eta)=I\Xi_{\mathcal{D}}Y$ have degree $L$, and let $r_{n,k}^{X,\mathcal{D}}$ be the coefficients in the $1+2L$-term constant-coefficient recurrence for the multi-indexed Me…

Let $P_{\mathcal{D},n}(\eta)$ denote the case-(1) multi-indexed Meixner–Pollaczek and continuous Hahn polynomials associated with an index set $\mathcal{D}$, where the denominator…