16 problems

The partition identity conjecture. For all integers $n,r,s e 0$, the following equivalent identities hold:

For $l,m\in\mathbb{N}$, define the indicator $[A]$ of an assertion $A$ by … Here $\lceil\cdot\rceil$ denotes the ceiling function, and binomial coefficients are defined for arbitra…

Polynomial power-compositions determinant conjecture. For any positive integers $n$ and $k$,

Shifted G-determinant identities. For all $n\geq3$, the three chains of identities displayed in the source hold:

Determinant evaluation conjecture. The determinant satisfies

Closed-form sum conjecture. One has

Let $l$ be a positive integer, and let $P(x)$ be a real polynomial of degree $n$ with leading coefficient $a_n$. Scaled generalized Tepper identity. One has … The preceding theorem…

Alternating Pascal-row identity. This inner product is equal to $n!$.

Cyclic q-binomial conjecture. For every integer $j$, this expression is a Laurent polynomial in $q$, and it has non-negative integer coefficients when $0\leqslant j\leqslant m$. Th…

Let $B_{n,k}:=\frac{k}{n}{2n\choose n-k}$ for $n\geqslant k\geqslant 0$, and let $r=\min\{n,m\}$ and $s=\max\{n,m\}$. Miana–Ohtsuka–Romero's conjecture. For all positive integers…

Let $a,b,c$ be non-negative integers and, for $r\in\mathbb{Z}^+$, define … Let $\mathcal{L}$ act on functions of $a,b,c$ by … where $E\cdot f(a,b,c)=f(a-1,b,c)$ and…

Let $n$ be a natural number, and let $f$ and $g$ be differentiable functions. Differentiable-functions identity. … The preceding identities establish special cases involving linear…

Nonintegrality conjecture. For any $r>s>1$, there is a positive integer $n$ such that $c_n^{(r,s)}$ is not an integer. Together with the finite integrality conjecture, this propose…

Finite integrality conjecture. For any $s>1$ and $n\geqslant 0$, there is an integer $r>s$ such that all the numbers $c_k^{(r,s)}$ for $0\leqslant k\leqslant n$ are integers. This…

Let $bc$ be an indeterminate. Define the rising factorial $(a)_m=a(a+1)\cdots(a+m-1)$, with $(a)_0=1$, and let $C(n)$, $F(n)$, and $G(n)$ be the sequences defined by the displayed…

Let $T_{2k}(b,1)$ denote the Lucas polynomial appearing in the identities (IV1)–(IV18). Consider identities of the form … where $a,d,m\in\mathbb Z$, $b\in\{1,3,4,\ldots\}$, $d>0$,…