Related papers: Krawtchouk polynomials, the Lie algebra $\mathfrak…

Fix an algebraically closed field $\mathbb{F}$ and an integer $n \geq 1$. Let $\text{Mat}_n(\mathbb{F})$ denote the $\mathbb{F}$-algebra consisting of the $n \times n$ matrices that have all entries in $\mathbb{F}$. We consider a pair of…

The dual Hahn polynomials $\{u_i(x)\}_{i=0}^d$ are a family of discrete orthogonal polynomials involving two real parameters $r$ and $s$. Let $L,L^*$ denote the corresponding Leonard pair. Assume that $r\not=0$ and $r+s=0$. We show that…

Eigenvalue problems on irreducible $\mathfrak{su}(2)$ modules and their adjoints are considered in the Bargmann, Barut-Girardello and finite difference models. The biorthogonality relations that arise between the corresponding generating…

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy both conditions below: (i) There exists a basis for $V$…

Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\to V$ and $A^*: V\to V$ that satisfy (i) and (ii) below. (i) There exists a basis for $V$ with respect to which the…

It is well-known that Leonard pairs have a close connection with bispectral orthogonal polynomials of the Askey scheme. In this paper, we introduce the notion of a Leonard trio $(V,\oV,Z)$, an algebraic structure extending Leonard pairs,…

Fix an algebraically closed field $\F$ and an integer $d \geq 3$. Let $V$ be a vector space over $\F$ with dimension $d+1$. A Leonard pair on $V$ is a pair of diagonalizable linear transformations $A: V \to V$ and $A^* : V \to V$, each…

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A : V \to V$ and $A^* : V \to V$ that satisfy (i) and (ii) below: (i) There exists a basis for…

Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy (i), (ii) below: (i) There exists a basis for $V$…

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. By a Leonard pair on $V$ we mean an ordered pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following two…

A Leonard pair is an ordered pair of diagonalizable linear maps on a finite-dimensional vector space, that each act on an eigenbasis for the other one in an irreducible tridiagonal fashion. In the present paper we consider a type of Leonard…

Krawtchouk's polynomials occur classically as orthogonal polynomials with respect to the binomial distribution. They may be also expressed in the form of matrices, that emerge as arrays of the values that the polynomials take. The algebraic…

We introduce a linear algebraic object called a bidiagonal pair. Roughly speaking, a bidiagonal pair is a pair of diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the…

Recently Terwilliger and the present author found a presentation for the three-point $\mathfrak{sl}_2$ loop algebra via generators and relations. To obtain this presentation we defined a Lie algebra $\boxtimes$ by generators and relations…

Krawtchouk polynomials play an important role in coding theory and are also useful in graph theory and number theory. Although the basic properties of these polynomials are to some extent known, there is, to my knowledge, no detailed…

Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and…

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (i), (ii) below. (i) There exists a…

Let $V$ denote a vector space over C with finite positive dimension. By a {\em Leonard triple} on $V$ we mean an ordered triple of linear operators on $V$ such that for each of these operators there exists a basis of $V$ with respect to…

Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\rightarrow V$ and $A^*: V\rightarrow V$ that satisfy (i) and (ii) below. (i) There exists a basis for $V$ with respect…

Let $\mathbb{K}$ denote an algebraically closed field. Let $V$ denote a vector space over $\mathbb{K}$ with finite positive dimension. By a Leonard triple on $V$ we mean an ordered triple of linear transformations in ${\rm End}(V)$ such…