25 problems

Let $q$ be a formal variable, and for integers $a_1,\dots,a_r$ define … For $i,j\geq 0$, let $(q;q)_i$ and $(q^3;q^3)_j$ denote the corresponding finite $q$-Pochhammer symbols. And…

Let $F_q^{\mathrm{E}_6}(L)$ denote the finitized fermionic polynomial. For $L\in\mathbb{Z}_{\geq0}$, define … … where the $q$-multinomial is understood to vanish unless its lower e…

Let $F_q^{\mathrm{D}_{n-1}}(L)$ denote the finitized fermionic polynomial and let $B_q^{\mathrm{D}_{n-1}}(L)$ denote the normalized bosonic polynomial defined from the…

Let $F_q^{\mathrm{A}_{n-1}}(L)$ denote the finitized fermionic polynomial and let $B_q^{\mathrm{A}_{n-1}}(L)$ denote the normalized bosonic polynomial defined from the…

Fix $r\geq 3$ and consider the pair $(T_1,D_r)$, with $A(T_1,D_r)=C(D_r)^{-1}$. Let $B$ range over vectors of the appropriate dimension, and let the associated generalized Nahm sum…

Let $q$ be a formal variable, and for $n\geq0$ let $(q^4;q^4)_n$ and $(q^m;q^m)_\infty$ denote the usual finite and infinite $q$-Pochhammer products; products with several argument…

Andrews–Uncu's conjecture.

Kang and Park's conjecture. For all positive integers $n,d\geq 1$,

For $m\geq 1$, let $R^{(2,2m)}$ denote the even plane curve singularity and let $\widehat{\mathit{NZ}}_{R^{(2,2m)}}$ be its completed numerator series. Rogers–Ramanujan-type conjec…

For $|q|<1$, let $(a;q)_n$ denote the $q$-Pochhammer symbol, defined by $(a;q)_n=\prod_{k=0}^{n-1}(1-aq^k)$ and $(a;q)_\infty=\prod_{k=0}^{\infty}(1-aq^k)$; for multiple parameters…

Let $a,b,k$ be integers such that $0\leqslant a,b\leqslant k$, let $\tau\in\{-1,0,1\}$, and set … For weakly decreasing nonnegative integer sequences…

Duncan–Khunger–Swisher–Tamura generalized conjecture. For $d,a,n\geq1$ with $1\leq a\leq d+2$,

Duncan–Khunger–Swisher–Tamura conjecture. For all $d,n\geq 1$,

Let $k,s$ be integers with $k\geqslant2$ and $1\leqslant s\leqslant k$, and let $(q)_n$, Gaussian $q$-binomial coefficients, and $\theta$ have their standard meanings. The…

Non-negative coefficient conjecture. For any profile $c=(c_1,c_2,c_3)$, the polynomial $P_{n,c}(q)$ has non-negative coefficients. Moreover, if $\ell\not\equiv 0\pmod{3}$, then

Let … where the polynomials satisfy $P_0(q)=1$, $P_1(q)=q+1$, $P_2(q)=q^3+q^2+q+1$, and $P_n(q)=(1+q^n)P_{n-1}(q)+qP_{n-2}(q)-qP_{n-3}(q)$ for $n\geq3$. Let $\mathrm T_1(N,M;q)$ de…

Let $P_n(q)$ be the polynomials defined by the recurrence in the source, with $P_0(q)=1$, $P_1(q)=q^2+1$, $P_2(q)=q^6+q^4+q^2+1$, and, for the displayed initial values, degree…

Three companion identities. These identities vary the linear term in the exponent of $q$ in a preceding companion identity and extend conjectural Rogers–Ramanujan-type identities a…

Let $q$ and $x$ be formal variables, and for each fixed integer $m\geq1$ let $n_{m,k}$ be positive integers indexed by finitely many $k\in I_m$. Refined Rogers–Ramanujan conjecture…

Let $\tau\in\mathbb{Z}$ be a framing, let $Z^{(\mathbb{C}^3,\mathcal{D}_\tau)}(\mathbf{x};q)$ be the framed open-string partition function, and let $f_n^\tau(q)$ be the coefficient…

A conformal field theory is considered that arises as the fixed point of a restricted solid-on-solid (RSOS) lattice model, and its characters are the associated conformal field the…

Let … where $(q;q)_k=\prod_{j=1}^{k}(1-q^j)$, with $(q;q)_0=1$. The second Rogers–Ramanujan identity. One has … This is one of the classical Rogers–Ramanujan identities, relating a…

Let $m,n\geq 1$. Let $h_{\lambda}^{(m)}(w,z;q)$ denote the modified Rogers–Szegő factor defined earlier, let $P'_{\lambda}$ be the modified Hall–Littlewood polynomial, and let…

For $1\leq a,b\leq N-1$, let $C_{ab}$ be the Cartan integers of the Lie algebra $\operatorname{A}_{N-1}$, with $C_{aa}=2$, $C_{a,a\pm1}=-1$, and $C_{ab}=0$ otherwise. For…

The Rogers–Ramanujan identities concern the graded structures of standard modules for affine Lie algebras. For $k>1$, higher-level standard $\widehat{\mathfrak{sl}}(2)$-modules can…