Bispectral commuting difference operators for multivariable Askey-Wilson polynomials

An algebraic description of the bispectrality of the biorthogonal rational functions of Hahn type

The biorthogonal rational functions of the ${_3}F_2$ type on the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials. These properties are described by three difference operators $X,Y,Z$ which are tridiagonal with respect to three distinct bases of the relevant finite-dimensional space. The pairwise commutators of the operators $X,Y,Z$ generate a quadratic algebra which is akin to the algebras of Askey-Wilson type attached to hypergeometric polynomials.

A $q$-generalization of the para-Racah polynomials

New bispectral orthogonal polynomials are obtained from an unconventional truncation of the Askey-Wilson polynomials. In the limit $q \to 1$, they reduce to the para-Racah polynomials which are orthogonal with respect to a quadratic bi-lattice. The three term recurrence relation and q-difference equation are obtained through limits of those of the Askey-Wilson polynomials. An explicit expression in terms of hypergeometric series and the orthogonality relation are provided. A $q$-generalization of the para-Krawtchouk polynomials is obtained as a special case. Connections with the $q$-Racah and …

Askey-Wilson Type Functions, With Bound States

The two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22], are slightly modified so as to make it transparent that these functions satisfy a beautiful symmetry property. It essentially means that the geometric and the spectral parameters are interchangeable in these functions. We call the resulting functions the Askey-Wilson functions. Then, we show that by adding bound states (with arbitrary weights) at specific points outside of the continuous spectrum of some instances of the Askey-Wilson diffe…

q-Hahn型有理函数的双谱性与双正交性

Bispectrality and biorthogonality of the rational functions of $q$-Hahn type

本文构造了一族关于q-超几何分布双正交的有理函数，证明三个q-差分算子X、Y、Z可类比正交多项式的双谱算子，并揭示其生成代数与Askey–Wilson型代数的联系及其与Wilson ${}_{10}ϕ_9$有理函数的关系。

q二次格上的伯恩斯坦型基与Askey–Wilson/q-Racah连接系数

Bernstein-type bases on $q$-quadratic lattices and Askey--Wilson\slash $q$-Racah connection coefficients

该研究在承载Askey–Wilson多项式的q二次格上拆分了线性格伯恩斯坦基的双重角色，确定了广义幂基与Askey–Wilson多项式的q-Racah型连接系数，并通过翼型案例阐释了正交机制。

Multivariable q-Racah polynomials

The Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials is studied for parameters satisfying a truncation condition such that the orthogonality measure becomes discrete with support on a finite grid. For this parameter regime the polynomials may be seen as a multivariable counterpart of the (one-variable) $q$-Racah polynomials. We present the discrete orthogonality measure, expressions for the normalization constants converting the polynomials into an orthonormal system (in terms of the normalization constant for the unit polynomial), and we discuss the limit $q\…

Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III

In a previous paper, we presented conjectures of the recurrence relations with constant coefficients for the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. In this paper we present a proof for the Laguerre and Jacobi cases. Their bispectral properties are also discussed, which give a method to obtain the coefficients of the recurrence relations explicitly. This paper extends to the Laguerre and Jacobi cases the bispectral techniques recently introduced by Gómez-Ullate et al. to derive explicit expressions for the coefficients of the recurrence relation…

Properties of some families of hypergeometric orthogonal polynomials in several variables

Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable Wilson, continuous Hahn and Jacobi type polynomials, respectively. For each class of polynomials we provide systems of difference (or differential) equations, recurrence relations, and expressions for the norms of the polynomials in terms of the norm of the constant polynomial.

对称函数代数中的q-Askey概型元素

Elements of the q-Askey scheme in the algebra of symmetric functions

本研究在对称函数代数中构造多变量q-Racah对称函数，推导其可退化为big q-Jacobi等三类对称函数，证明四类函数均关于无穷点构形空间上的测度正交，且测度间的极限转移与函数退化一致，丰富了q-Askey概型的多变量对称函数理论。

Symmetric abstract hypergeometric polynomials

Consider an abstract operator $L$ which acts on monomials $x^n$ according to $L x^n= λ_n x^n + ν_n x^{n-2}$ for $λ_n$ and $ν_n$ some coefficients. Let $P_n(x)$ be eigenpolynomials of degree $n$ of $L$: $L P_n(x) = λ_n P_n(x)$. A classification of all the cases for which the polynomials $P_n(x)$ are orthogonal is provided. A general derivation of the algebras explaining the bispectrality of the polynomials is given. The resulting algebras prove to be central extensions of the Askey-Wilson algebra and its degenerate cases.

The generic quantum superintegrable system on the sphere and Racah operators

We consider the generic quantum superintegrable system on the $d$-sphere with potential $V(y)=\sum_{k=1}^{d+1}\frac{b_k}{y_k^2}$, where $b_k$ are parameters. Appropriately normalized, the symmetry operators for the Hamiltonian define a representation of the Kohno-Drinfeld Lie algebra on the space of polynomials orthogonal with respect to the Dirichlet distribution. The Gaudin subalgebras generated by Jucys-Murphy elements are diagonalized by families of Jacobi polynomials in $d$ variables on the simplex. We define a set of generators for the symmetry algebra and we prove that their action on t…

On the diagonalization of difference Calogero-Sutherland systems

We discuss the simultaneous diagonalization of a family of commuting difference operators by Koornwinder's multivariable generalization of the Askey-Wilson polynomials. The operators constitute a complete set of quantum integrals for a difference version of the $n$-particle quantum Calogero-Sutherland system related to the root system $BC_n$.