23 problems

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

Weyl-antisymmetry conjecture. This polynomial satisfies

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…

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

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…

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…

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

Let $n \equiv 1\pmod{4}$ 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 $n$ be a positive integer with $n\equiv3\pmod4$, let $[n]=(1-q^n)/(1-q)$, let $(a;q)_k$ be the $q$-shifted factorial, and let $(a_1,\ldots,a_r;q)_k=\prod_{i=1}^r(a_i;q)_k$. Set…

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.…

The conjecture. Then

The conjecture. The truncated series should satisfy

Let $n$ be a positive integer with $n\equiv 3\pmod{4}$, let $r$ be any positive integer, and let $\Phi_n(q)$ denote the $n$th cyclotomic polynomial. Then the conjectured q-congruen…

Let $n$ and $r$ be nonnegative integers with $r\leqslant n$, and let $q,x$ be parameters for which the displayed expressions are defined. The q-Clausen-type conjecture. … The text…

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 $n\geq 0$, and let the basic hypergeometric series ${}_4\phi_3$ and the $q$-shifted factorials be defined by … where…