31 problems

Andrews–Merca–Guo–Zeng conjecture. This coefficient is non-negative. This conjecture generalizes non-negativity results for averaged truncations of Euler's pentagonal number theore…

For any complex numbers $x$ and $q$, define the $q$-shifted factorial by … and, for $n\in\mathbb{Z}^{+}$, … Write $(x_1,x_2,\dots,x_m;q)_n=(x_1;q)_n\cdots(x_m;q)_n$, let…

Let $n$ be a positive integer, let $x$ be a variable, and let $a$, $b$, $c$, $d$, and $q$ be parameters. Let $p_n(x;a,b,c,d;q)$ denote the Askey–Wilson polynomial. Quadratic relati…

Let $n$ be a positive integer, let $g$ and $ell$ be parameters, and let $\lambda=(\lambda_1,\ldots,\lambda_n)$. Write $(y;q)_m$ for the $q$-Pochhammer symbol and…

Three-variable basic hypergeometric conjecture. The series

Let $u_1,\ldots,u_n$ be indeterminates, let $\theta=(\theta_{i,j})\in\mathsf{M}^{(n)}$, and let $R_{ij}$ denote the raising operator. For a partition…

Weyl-antisymmetry conjecture. This polynomial satisfies

Commutativity conjecture. For general $n\geq 2$, one has

Commutativity conjecture. The family of operators $I(\alpha)$ is commutative on ${\cal F}_n$; equivalently, $[I(\alpha),I(\beta)]=0$.

Let $n$ be a positive integer, let ${\cal F}_n$ be the space of formal power series of degree zero in variables $\zeta_1,\ldots,\zeta_n$ corresponding to the positive cone of the…

Let $A$, $B$, and $C$ be noncommutative parameters in a Banach algebra, with $Q$ commuting with each of $A$, $B$, and $C$. Assume that $BC^{-1}A$ commutes with each of $A$, $B$, an…

A bilateral sum is a bilateral basic hypergeometric summation, and Cauchy's method is the procedure of deriving a bilateral identity from a terminating identity by an appropriate l…

Warnaar's conjecture. These two expressions are equal. This conjectural identity would extend the hierarchy of transformations between multiple basic hypergeometric series of diffe…

Let $(k,l,m,n)=(0,l,l,0)$ with $l\geq 4$, and let $\zeta_l$ be a primitive $l$-th root of unity. Consider the three-term relation … . Three-term relation conjecture. The three-term…

Let $n$ be a positive odd integer, and let $c_q(k)$ be the sequence defined in Theorem 8 by … Let $[n]$ denote the $q$-integer and let $\Phi_n(q)$ denote the $n$th cyclotomic polyn…

Let $n$ be a positive odd integer, and let $c_q(k)$ be the sequence defined in Theorem 7. Let $[n]$ denote the $q$-integer and let $\Phi_n(q)$ denote the $n$th cyclotomic polynomia…

Let $d$, $n$, and $r$ be positive integers with $\gcd(d,n)=1$ and $n$ odd. For $k\geq0$, define … where $(x;q)_k$ is the $q$-shifted factorial, and let $\Phi_n(q)$ denote the $n$th…

Denominator conjecture. For every $n\geq 0$,

Let $n equiv 1pmod{4}$ be a positive integer. Define the $q$-integer $[m]=(1-q^m)/(1-q)$, the $q$-shifted factorial $(a;q)k=prod{j=0}^{k-1}(1-aq^j)$, and let $Phin(q)$ denote the…

Let $s=(s_1,\ldots,s_n)$, let $\theta=\{\theta_{ij}\}_{1\leqslant i<j\leqslant n}$ be nonnegative integers, and define … The branching coefficients $e_n(s;\theta;q,t)$ are defined…

Vanishing integral conjecture. The integral $I_{\lambda}(a,b;q,t,t^n)$ vanishes unless $\operatorname{2\text{-}core}(\lambda)=0$. Moreover, the two specializations satisfy

The $(\mathrm{C}_n,\mathrm{B}_n)$ branching-rule conjecture. The two displayed identities hold.

Second coefficient-denominator conjecture. For every $n$, there is a polynomial $P'_n(a,b,c)$ in $a,b,c,q$ with integer coefficients such that

Let $(a;q)_k=\prod_{j=0}^{k-1}(1-aq^j)$ be the $q$-shifted factorial, let $\Phi_n(q)$ be the $n$th cyclotomic polynomial, and let $n>1$ be odd and $r\geqslant 1$. Guo–Zudilin's hal…

Let $d\geqslant4$ and let $n>1$ satisfy $n\equiv-1\pmod d$. Let $(a;q)_k$ denote the $q$-shifted factorial and let $\Phi_n(q)$ be the $n$th cyclotomic polynomial. The q-congruence.…