---
abstract: |
    Let $V$ denote a vector space with finite positive dimension, and let $(A,A^*)$ denote a Leonard pair on $V$. As is known, the linear transformations $A$, $A^*$ satisfy the Askey-Wilson relations $$\begin{aligned}
    A^2 A^*-\beta A A^*\!A+A^*\!A^2-\gamma\left( A A^*\!+\!A^*\!A
    \right)-\varrho\,A^* {&\!\!\!=\!\!\!&}\gamma^*\!A^2+\omega A+\eta\,I,\\
    A^*{}^2\!A-\beta A^*\!AA^*\!+AA^*{}^2\!-\gamma^*\!\left(A^*\!A\!+\!A
    A^*\right)-\varrho^*\!A {&\!\!\!=\!\!\!&}\gamma A^*{}^2\!+\omega A^*\!+\eta^*I,\end{aligned}$$ for some scalars $\beta,\gamma,\gamma^*,\varrho,\varrho^*,\omega,\eta,\eta^*$. The scalar sequence is unique if the dimension of $V$ is at least $4$.

    If $c,c^*\!,t,t^*\!$ are scalars and $t,t^*\!$ are not zero, then $(tA+c,t^*\!A^*\!+c^*)$ is a Leonard pair on $V$ as well. These affine transformations can be used to bring the Leonard pair or its Askey-Wilson relations into a convenient form. This paper presents convenient normalizations of Leonard pairs by the affine transformations, and exhibits explicit Askey-Wilson relations satisfied by them.\
    [**AMS 2000 MSC Classification**]{}: 05E35, 33D45, 33C45.\
    [**Keywords**]{}: Leonard pairs, Askey-Wilson relations.\
    [**E-mail address**]{}: [vidunas@math.kyushu-u.ac.jp]{}.\
author:
- |
    Raimundas Vidūnas[^1]\
    *Kyushu University*
bibliography:
- '../terwilliger.bib'
- '../../hypergeometric.bib'
title: 'Normalized Leonard pairs and Askey-Wilson relations'
---

Introduction
============

Throughout the paper, ${{\Bbb K}}$ denotes an algebraically closed field. Apart from one remark, we assume the characteristic of ${{\Bbb K}}$ is not equal to 2.

Recall that a tridiagonal matrix is a square matrix which has 