Commun. Korean Math. Soc. 2019; 34(2): 401-414
Online first article March 15, 2019 Printed April 1, 2019
https://doi.org/10.4134/CKMS.c180181
Copyright © The Korean Mathematical Society.
Hasan Alnajjar
The University of Jordan
Let $\mathcal{F}$ denote an algebraically closed field with characteristic not two. Fix an integer $d\geq 3$, let $\mathrm{Mat}_{d+1}(\mathcal{F})$ denote the $\mathcal{F}$-algebra of $(d+1)\times(d+1)$ matrices with entries in $\mathcal{F}$. An ordered pair of matrices $A$, $A^*$ in $\mathrm{Mat}_{d+1}(\mathcal{F})$ is said to be LB-TD form whenever $A$ is lower bidiagonal with subdiagonal entries all $1$ and $A^*$ is irreducible tridiagonal. Let $A$, $A^*$ be a Leonard pair in $\mathrm{Mat}_{d+1}(\mathcal{F})$ with fundamental parameter $\beta =2$, with this assumption there are four families of Leonard pairs, Racah, Hahn, dual Hahn, Krawtchouk type. In this paper we show from these four families only Racah and Krawtchouk have LB-TD form.
Keywords: Leonard pair, Askey-Wilson relation, Racah polynomial, Hahn polynomial, dual Hahn polynomial, Krawtchouk polynomial
MSC numbers: 05E10, 05E30, 33C45, 33D45
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