相关论文: Some Systems of Multivariable Orthogonal Askey-Wil…
Multivariable generalizations of the continuous Hahn and Wilson polynomials are introduced as eigenfunctions of rational Ruijsenaars type difference systems with an external field.
This paper surveys eight classes of polynomials associated with $A$-type and $BC$-type root systems: Jack, Jacobi, Macdonald and Koornwinder polynomials and interpolation (or shifted) Jack and Macdonald polynomials and their $BC$-type…
In this paper, by introducing new matrix operations and using a specific inverse relation, we establish the dual forms of the orthogonality relations for some well-known discrete and continuous $q$-orthogonal polynomials from the…
The Askey--Wilson integral is very important in the theory of orthogonal polynomials. Liu's integral is a generalization of the Askey--Wilson integral with many parameters. With the help of the series rearrangement method, we give the…
It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a…
An overview of the basic results on Macdonald(-Koornwinder) polynomials and double affine Hecke algebras is given. We develop the theory in such a way that it naturally encompasses all known cases. Among the basic properties of the…
We develop a unified construction of matrix-valued orthogonal polynomials associated with discrete weights, yielding bispectral sequences as eigenfunctions of second-order difference operators. This general framework extends the discrete…
The strict relation between some class of multiboson hamiltonian systems and the corresponding class of orthogonal polynomials is established. The correspondence is used effectively to integrate the systems. As an explicit example we…
Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme. We show how asymptotic representations can be derived by using the generating functions of the polynomials. For example, we discuss the…
We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials.…
In this paper we present an addition to Askey's scheme of q-hypergeometric orthogonal polynomials involving classes of q-special functions which do not consist of polynomials only. The special functions are q-analogues of the Jacobi and…
The multivariate quantum $q$-Krawtchouk polynomials are shown to arise as matrix elements of "$q$-rotations" acting on the state vectors of many $q$-oscillators. The focus is put on the two-variable case. The algebraic interpretation is…
In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials…
A LG-WKB and Turning point theory is developed for three term recurrence formulas associated with monotonic recurrence coefficients. This is used to find strong asymptotics for certain classical orthogonal polynomials including Wilson…
We study skew-orthogonal polynomials with respect to the weight function $\exp[-2V(x)]$, with $V(x)=\sum_{K=1}^{2d}(u_{K}/{K})x^{K}$, $u_{2d} > 0$, $d > 0$. A finite subsequence of such skew-orthogonal polynomials arising in the study of…
The exceptional Racah and q-Racah polynomials are constructed. Together with the exceptional Laguerre, Jacobi, Wilson and Askey-Wilson polynomials discovered by the present authors in 2009, they exhaust the generic exceptional orthogonal…
A $\mathbb{D}$-semi-classical weight is one which satisfies a particular linear, first order homogeneous equation in a divided-difference operator $\mathbb{D}$. It is known that the system of polynomials, orthogonal with respect to this…
This work explores classical discrete multiple orthogonal polynomials, including Hahn, Meixner of the first and second kinds, Kravchuk, and Charlier polynomials, with an arbitrary number of weights. Explicit expressions for the recursion…
A limiting property of the nearest-neighbor recurrence coefficients for multiple orthogonal polynomials from a Nevai class is investigated. Namely, assuming that the nearest-neighbor coefficients have a limit along rays of the lattice, we…
We obtain weight functions associated with $q$-linear and $q$-quadratic lattices that yield discrete orthogonality with respect to a quasi-definite moment functional for the Askey-Wilson polynomials and all the polynomial sequences in the…