The end-parameters of a Leonard pair

Abstract

Fix an algebraically closed field $\F$ and an integer $d\ge 3$. 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 $A^{\ast }:V\to V$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. There is an object related to a Leonard pair called a Leonard system. It is known that a Leonard system is determined up to isomorphism by a sequence of scalars $(\{þ_i\}_{i=0}^d, \{þ^*_i\}_{i=0}^d, \{\vphi_i\}_{i=1}^d, \{ϕ_i\}_{i=1}^d)$, called its parameter array. The scalars $\{{þ}_i{\}}_{i=0}^d$ (resp.\ $\{{þ}_i^{\ast }{\}}_{i=0}^d$) are mutually distinct, and the expressions $({þ}_{i-2}-{þ}_{i+1})/({þ}_{i-1}-{þ}_i)$, $({þ}_{i-2}^{\ast }-{þ}_{i+1}^{\ast })/({þ}_{i-1}^{\ast }-{þ}_i^{\ast })$ are equal and independent of $i$ for $2\le i\le d-1$. Write this common value as $\beta +1$. In the present paper, we consider the "end-parameters" ${þ}_0$, ${þ}_d$, ${þ}_0^{\ast }$, ${þ}_d^{\ast }$, $\vphi_1$, $\vphi_d$, ${\varphi }_1$, ${\varphi }_d$ of the parameter array. We show that a Leonard system is determined up to isomorphism by the end-parameters and $\beta$. We display a relation between the end-parameters and $\beta$. Using this relation, we show that there are up to inverse at most $\lfloor (d-1)/2\rfloor$ Leonard systems that have specified end-parameters. The upper bound $\lfloor (d-1)/2\rfloor$ is best possible.

References in corpus (4)

• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an algebraic approach to the Askey scheme of orthogonal polynomials
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array
• Affine transformations of a Leonard pair
• Some trace formulae involving the split sequences of a Leonard pair

Cited by in corpus (1)

• Leonard pairs having specified end-entries