113 problems

Let $G(x,y,t)$ be the full generating function for Gessel walks with step set $\\{\nearrow,\swarrow,\leftarrow,\rightarrow\\}$, and let $G(0,0,t)$ be the generating function of Ges…

A differential equation is globally nilpotent if its $p$-curvature is nilpotent for almost all primes $p$. A Gaussian hypergeometric equation is a differential equation associated…

Special-value conjecture.

Iterated-derivative toric-residue conjecture. A rational $A$-hypergeometric function has an iterated derivative that is a linear combination of toric residues associated with facia…

A Feynman integral is an integral arising from a perturbative quantum field theory amplitude, and a hypergeometric function in the general sense is a function belonging to the broa…

Let $p$ and $q$ be integers with $p<q$, let ${\mathbf b}=(b_1,\ldots,b_q)$ satisfy $b_i>0$, and let ${\mathbf a}=(a_1,\ldots,a_p)$ satisfy $a_i\geq b_i$ for $1\leq i\leq p$. The ge…

Let $j(\tau)$ be the Klein $j$-function, and let $i$ and $\rho=e^{\pi i/3}$ be the elliptic points of the full modular group. Consider the Taylor series expansions of the inverse o…

Let $\Delta_k$ denote the $k$-simplex, and let … where $k_0\geq\cdots\geq k_r$. The configuration $A$ is GKZ-rational when its associated GKZ hypergeometric system has the rational…

Let $A$ be a configuration, let $D_A$ denote its discriminant, and let a rational $A$-hypergeometric function be a rational solution associated with $A$. A resultant…

Let $A_0,A_1,\dots,A_r$ be vector configurations in $\mathbb{Z}^r$, and let $A$ be their Cayley configuration … where $e_0,\dots,e_r$ is the standard basis of $\mathbb{Z}^{r+1}$. T…

A rational hypergeometric function is a function associated with a toric configuration whose denominator may involve discriminants of that configuration. A resultant is a special d…

The level-zero qKZ equation has hypergeometric solutions constructed by the method introduced for the $\mu=0$ case, and these solutions can satisfy linear relations. Completeness c…

Conjecture 2. Under these assumptions, the inequality stated immediately after this parameter condition holds.

Conjecture 1. Theorem LowerGeneral is true for all $\sigma>0$ and

Connection-formula conjecture. Repeated application of the algorithm for degenerated numerator parameters and of contiguity relations yields a connection formula for any degenerate…

When the highest weights are integral, the qKZB and Macdonald–Ruijsenaars operators may have singularities on the lattice, so their modified versions are defined by removing the si…

Larcombe's conjecture. Peter Larcombe conjectured that

Let $n$ be the dimension parameter, let $\rho=(n-1)/2$, and define … where $F(a,b;c;z)$ is the hypergeometric function and $k$ is complex. The zero-free region conjecture. For ever…

Let $n$ be the dimension parameter, let $\rho=(n-1)/2$, and let $F(a,b;c;z)$ denote the hypergeometric function. For the function $F_k(z)=F(k,-\rho;k+n/2;z)$, with complex variable…

The paper considers the coefficients in the left-hand side of formula (12), expressed using the quantities $G_0(k,n,s;1)$. Coefficient and Ohno–Zagier conjecture. All coefficients…

The rational quantized Knizhnik–Zamolodchikov difference equations have hypergeometric solutions, and the corresponding dynamical differential equations are defined for the same se…

Let $n$, $A$, $B$, and $C$ satisfy the conditions imposed above, and let ${\bf p}_{0,C,n}(z)$ and ${\bf p}_{l,n}(z)$ be the coefficient functions occurring in the specialized hyper…

A KZ equation or dynamical equation is considered whose solutions take values in a one-dimensional space. Gamma-function formula conjecture. If the space of values of such an equat…

Generalized hypergeometric expression conjecture. For a given $\ell\geq 0$, the components of the functions $H(t)$ are expressible in terms of generalized hypergeometric functions…

Hypergeometric-form conjecture. For a given $\ell\geq 0$, the spherical functions corresponding to $(\ell,n)$ have components expressible in terms of generalized hypergeometric fun…