Related papers: Bispectral Jacobi type polynomials
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 class of three-diagonal Jacobi matrix with exponentially increasing elements is considered. Under some assumptions the matrix corresponds to unbounded self-adjoint operator in the weighted space. The weight depends on elements of the…
We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-$D$ Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the…
We study the properties and asymptotics of the Jacobi matrices associated with equilibrium measures of the weakly equilibrium Cantor sets. These family of Cantor sets were defined and different aspects of orthogonal polynomials on them were…
Representation of analytic functions as convergent series in Jacobi polynomials $P_n^{(a,b)}$ is reformulated using a unified approach for almost all complex $a, b$. The coefficients of the series are given as usual integrals in the…
The possibility for the Jacobi equation to admit in some cases general solutions that are polynomials has been recently highlighted by Calogero and Yi, who termed them para-Jacobi polynomials. Such polynomials are used here to build seed…
A one-parameter family of operators that have the complementary Bannai-Ito (CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the kernel partners of the Bannai-Ito polynomials and also correspond to a $q\rightarrow-1$…
Orthogonal polynomials for a family of weight functions on $[-1,1]^2$, $$ \CW_{\a,\b,\g}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} (1-x^2)^\g(1-y^2)^{\g}, $$ are studied and shown to be related to the Koornwinder polynomials defined on the region…
Banded bounded matrices, which represent non normal operators, of oscillatory type that admit a positive bidiagonal factorization are considered. To motivate the relevance of the oscillatory character the Favard theorem for Jacobi matrices…
Functions like the exponential, Chebyshev polynomials, and monomial symmetric polynomials are preeminent among all special functions. They have simple definitions and can be expressed using easily specified integers like n!. Families of…
We discuss computing with hierarchies of families of (potentially weighted) semiclassical Jacobi polynomials which arise in the construction of multivariate orthogonal polynomials. In particular, we outline how to build connection and…
A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse…
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…
It is established that the existence of non-isotropic vector field which Jacobi operator of maximal rank is an obstacle for the existence of non-trivial second-order symmetric parallel tensor field. In turns out that presence of such…
We construct new families of discrete vector orthogonal polynomials that have the property to be eigenfunctions of some difference operator. They are extensions of Charlier, Meixner and Kravchuk polynomial systems. The ideas behind our…
This manuscript contains a small portion of the algebraic theory of orthogonal polynomials developed by Maroni and their applicability to the study and characterization of the classical families, namely Hermite, Laguerre, Jacobi, and Bessel…
Jack polynomials in superspace, orthogonal with respect to a ``combinatorial'' scalar product, are constructed. They are shown to coincide with the Jack polynomials in superspace, orthogonal with respect to an ``analytical'' scalar product,…
We describe symmetric diffusion operators where the spectral decomposition is given through a family of orthogonal polynomials. In dimension one, this reduces to the case of Hermite, Laguerre and Jacobi polynomials. In higher dimension,…
Orthogonal polynomials $P_{n}(\lambda)$ are oscillating functions of $n$ as $n\to\infty$ for $\lambda$ in the absolutely continuous spectrum of the corresponding Jacobi operator $J$. We show that, irrespective of any specific assumptions on…
In this paper, we introduce Jacobi polynomial generalizations of several classical invariants in coding theory over finite fields, specifically, the higher and extended weight enumerators, and we establish explicit correspondences between…