Related papers: The rational Heun operator and Wilson biorthogonal…
We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For $\mathfrak{su}(2)$, this leads to the Heun-Krawtchouk algebra. The corresponding…
We construct a family of bilinear differential operators which satisfy certain gauge properties. These operators can be naturally associated with $q$-deformations of classical integrable hierarchies. In particular, we consider the case when…
In this paper, we study Hamiltonian operators which are sum of a first order operator and of a Poisson tensor, in two spatial independent variables. In particular, a complete classification of these operators is presented in two and three…
A finite family of $R_I$ polynomials is introduced and studied. It consists in a set of polynomials of $_{3}F_{2}$ form whose biorthogonality to an ensemble of rational functions is spelled out. These polynomials are shown to satisfy two…
The irreps $(SU(2),{\cal H},U)$ of SU(2) of dimension $(2S+1)^N$, i.e. operators acting on the space ${\cal H}={\cal H}_N={\bf C}^{(2S+1)^N}$ of $N$ identical particles with spin $S$, are described by Clebsch-Gordan decomposition into…
The Heun functions satisfy linear ordinary differential equations of second order with certain singularities in the complex plane. The first order derivatives of the Heun functions satisfy linear second order differential equations with one…
We define Hecke operators U_m that sift out every m-th Taylor series coefficient of a rational function in one variable, defined over the reals. We prove several structure theorems concerning the eigenfunctions of these Hecke operators,…
In resonance to a recent geometric framework proposed by Douglas and Yang, a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space is developed. By taking advantage of the refined…
We introduce an algebra $\mathcal H$ consisting of difference-reflection operators and multiplication operators that can be considered as a $q=1$ analogue of Sahi's double affine Hecke algebra related to the affine root system of type…
The 'relativistic' Heun equation is an 8-coupling difference equation that generalizes the 4-coupling Heun differential equation. It can be viewed as the time-independent Schr\"odinger equation for an analytic difference operator introduced…
We introduce the algebraic Heun operator associated to any bispectral pair of operators. We show that these operators are natural generalizations of the ordinary Heun operator. This leads to a simple construction of the operators commuting…
From some observations on the linear differential operators occurring in the Lattice Green function of the d-dimensional face centred and simple cubic lattices, and on the linear differential operators occurring in the n-particle…
Orthogonal polynomials with respect to the hypergeometric distribution on lattices in polyhedral domains in ${\mathbb R}^d$, which include hexagons in ${\mathbb R}^2$ and truncated tetrahedrons in ${\mathbb R}^3$, are defined and studied.…
Considering a differential operator of third order that does not increase the degree of polynomials, we analyse some properties of elements of the dual space of 2-orthogonal polynomial eigenfunctions. In two classes of such generic…
In this work, the Heun operator is written as an element in the universal enveloping algebra of the Lie algebra $\mathscr{G}=\mathscr{L}(G)$ of the Lie group $G=SL(2,\mathbb{C})$. The Green function and the spectral shift function of the…
We introduce the symmetric (respectively, non-symmetric) $\tau_{-\ell}-$hypergeometric functions associated with a root system of type $BC$ as joint eigenfunctions of a commutative algebra of differential (respectively,…
In the paper we deal with the Heun functions --- solutions of the Heun equation, which is the most general Fuchsian equation of second order with four regular singular points. Despite the increasing interest to the equation and numerous…
Motivated by Liu's recent work in \cite{Liu2022}. We shall reveal the essential feature of Hahn polynomials by presenting two new $q$-exponential operators. These lead us to use a systematic method to study identities involving Hahn…
We consider the exactly solvable quantum mechanical systems whose eigenfunctions are described by the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. Corresponding to the recurrence relations with…
We derive the most general families of differential operators of first and second degree semi-commuting with the differential operators of the Heun class. Among these families we classify all those families commuting with the Heun class. In…