The Heun–Askey–Wilson Algebra and the Heun Operator of Askey–Wilson Type

  title={The Heun–Askey–Wilson Algebra and the Heun Operator of Askey–Wilson Type},
  author={Pascal Baseilhac and Satoshi Tsujimoto and Luc Vinet and Alexei S. Zhedanov},
  journal={Annales Henri Poincar{\'e}},
The Heun-Askey-Wilson algebra is introduced through generators $\{\boX,\boW\}$ and relations. These relations can be understood as a, extension of the usual Askey-Wilson ones. A central element is given, and a canonical form of the Heun-Askey-Wilson algebra is presented. A homomorphism from the Heun-Askey-Wilson algebra to the Askey-Wilson one is identified. On the vector space of the polynomials in the variable $x=z+z^{-1}$, the Heun operator of Askey-Wilson type realizing $\boW$ can be… Expand
Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz
Abstract An operator of Heun-Askey-Wilson type is diagonalized within the framework of the algebraic Bethe ansatz using the theory of Leonard pairs. For different specializations and the genericExpand
Heun algebras of Lie type
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 theExpand
The rational Heun operator and Wilson biorthogonal functions
We consider the rational Heun operator defined as the most general second-order $q$-difference operator which sends any rational function of type $[(n-1)/n]$ to a rational function of typeExpand
A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$
Building on classical invariant theory, it is observed that the polarised traces generate the centraliser $Z_L(sl(N))$ of the diagonal embedding of $U(sl(N))$ in $U(sl(N))^{\otimes L}$. The paperExpand
Elliptic Racah polynomials
The Askey-Wilson polynomials [AW85] constitute a master family from which all other members listed in Askey’s celebrated scheme of (basic) hypergeometric orthogonal polynomials can be recovered viaExpand
Heun operator of Lie type and the modified algebraic Bethe ansatz
The generic Heun operator of Lie type is identified as a certain $BC$-Gaudin magnet Hamiltonian in a magnetic field. By using the modified algebraic Bethe ansatz introduced to diagonalize such GaudinExpand
The rational Sklyanin algebra and the Wilson and para-Racah polynomials
The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second order Heun operators on quadratic grids with no diagonalExpand
The missing label of $\mathfrak{su}_3$ and its symmetry
We present explicit formulas for the operators providing missing labels for the tensor product of two irreducible representations of su3. The result is seen as a particular representation of theExpand
Heun’s differential equation and its q-deformation
  • K. Takemura
  • Mathematics, Physics
  • 2019
The $q$-Heun equation is a $q$-difference analogue of Heun's differential equation. We review several solutions of Heun's differential equation and investigate polynomial-type solutions of $q$-HeunExpand
q-Heun equation and initial-value space of q-Painlev\'e equation
We show that the q-Heun equation and its variants appear in the linear q-difference equations associated to some q-Painlevé equations by considering the blow-up associated to their initial-valueExpand


Little and big q-Jacobi polynomials and the Askey–Wilson algebra
The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey–Wilson algebra. The operators that these polynomials respectively diagonalize areExpand
“Hidden symmetry” of Askey-Wilson polynomials
ConclusionsWe have shown that the Askey-Wilson polynomials of general form are generated by the algebra AW(3), which has a fairly simple structure and is the q-analog of a Lie algebra with threeExpand
Symmetry techniques for $q$-series: Askey-Wilson polynomials
We advocate the exploitation of symmetry (recurrence relation) techniques for the derivation of properties associated with families of basic hypergeometric functions, in analogy with the local LieExpand
Tridiagonalization and the Heun equation
It is shown that the tridiagonalization of the hypergeometric operator $L$ yields the generic Heun operator $M$. The algebra generated by the operators $L,M$ and $Z=[L,M]$ is quadratic and aExpand
Algebraic Heun Operator and Band-Time Limiting
We introduce the algebraic Heun operator associated to any bispectral pair of operators. This operator can be presented as a generic bilinear combination of these bispectral operators. We show thatExpand
The Heun operator of Hahn-type
The Heun-Hahn operator on the uniform grid is defined. This operator is shown to map polynomials of degree $n$ to polynomials of degree $n+1$, to be tridiagonal in bases made out of either PochhammerExpand
On q-Deformations of the Heun Equation
  • K. Takemura
  • Mathematics, Physics
  • Symmetry, Integrability and Geometry: Methods and Applications
  • 2018
The q-Heun equation and its variants were obtained by degenerations of Ruijsenaars-van Diejen operators with one particle. We investigate local properties of these equations. Especially weExpand
Mutual integrability, quadratic algebras, and dynamical symmetry
Abstract The concept of mutually integrable dynamical variables is proposed. This concept leads to the quadratic Askey-Wilson algebra QAW(3) which is the dynamical symmetry algebra for all problemsExpand
Spectral Analysis of Certain Schrodinger Operators
The J-matrix method is extended to difference and q-difference operators and is applied to several explicit differential, difference, q-difference and second order Askey{ Wilson type operators. TheExpand
Poincaré-Birkhoff-Witt deformations of Calabi-Yau algebras
Recently, Bocklandt proved a conjecture by Van den Bergh in its graded version, stating that a graded quiver algebra A (with relations) which is Calabi-Yau of dimension 3 is defined from aExpand