Related papers: Orthogonal polynomials and the deformed Jordan pla…
For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal…
We classify the finite dimensional irreducible representations of rectangular finite $W$-algebras, i.e., the finite $W$-algebras $U(\mathfrak{g}, e)$ where $\mathfrak{g}$ is a symplectic or orthogonal Lie algebra and $e \in \mathfrak{g}$ is…
It is shown that the finite-dimensional simple representations of the super Jordan plane $B$ are one-dimensional. The indecomposable representations of dimension $2$ and $3$ of $B$ are classified. Two families of indecomposable…
We introduce a ten-parameter ordinary linear differential equation of the second order with four singular points. Three of these are finite and regular whereas the fourth is irregular at infinity. We use the tridiagonal representation…
We develop a theory of Jacobi polynomials for parabolic subgroups of finite reflection groups that specializes to the cases studied by Heckman and Opdam in which the whole group and the trivial group are considered. For the intermediate…
A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\cal H}(X)$. This…
We derive orthogonality relations for discrete q-ultraspherical polynomials and their duals by means of operators of representations of the quantum algebra su_q(1,1). Spectra and eigenfunctions of these operators are found explicitly. These…
We construct an explicit diagonal \Delta_P on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks' projection P --> K and its factorization through J. We introduce the notion of a…
The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of…
In this article we prove that the elliptic, hyperbolic and nilpotent (or unipotent) additive (or multiplicative) Jordan components of an endomorphism $X$ (or an isomorphism $g$) of a finite dimensional vector space are given by polynomials…
Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions,…
We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A) via Fock space reformulations of their Kazhdan-Lusztig theories. As a consequence, the characters of finite-dimensional…
A general scheme for tridiagonalising differential, difference or q-difference operators using orthogonal polynomials is described. From the tridiagonal form the spectral decomposition can be described in terms of the orthogonality measure…
A joint algebraic interpretation of the biorthogonal Askey polynomials on the unit circle and of the orthogonal Jacobi polynomials is offered. It ties their bispectral properties to an algebra called the meta-Jacobi algebra $m\mathfrak{J}$.
We compute the graded rank of the cohomology of the hyperplane complement associated with a quaternionic reflection group, and observe that it factors into irreducible factors with positive integer coefficients. For an irreducible group,…
In this paper we study the orthogonality conditions satisfied by Jacobi polynomials $P_n^{(\alpha,\beta)}$ when the parameters $\alpha$ and $\beta$ are not necessarily $>-1$. We establish orthogonality on a generic closed contour on a…
A Bernstein type inequality is obtained for the Jacobi polynomials $P_n^{\alpha,\beta}(x)$, which is uniform for all degrees $n\ge0$, all real $\alpha,\beta\ge0$, and all values $x\in [-1,1]$. It provides uniform bounds on a complete set of…
For every family of orthogonal polynomials, we define a new realization of the Yangian of ${\mathfrak{gl}}_n$. Except in the case of Dickson polynomials, the new realizations do not satisfy the RTT relation. We obtain an analogue of the…
An algebraic interpretation of the one-variable quantum $q$-Krawtchouk polynomials is provided in the framework of the Schwinger realization of $\mathcal{U}_{q}(sl_{2})$ involving two independent $q$-oscillators. The polynomials are shown…
Let $\mathfrak{A}$ be a unital ring with a nontrivial idempotent. In this paper, it is shown that under certain conditions every multiplicative generalized Jordan $n$-derivation $\Delta:\mathfrak{A}\rightarrow\mathfrak{A}$ is additive. More…