$\omega$ are valid for $2\le i\le d-1$, the expressions for $\varrho$, $\varrho^*$ are valid for $1\le i\le d$, and the expressions for $\gamma,\gamma^*,\eta,\eta^*$ are valid for $1\le i\le
d-1$. The numbers $a_i,a_i^*$ are the diagonal entries in the tridiagonal forms of $A$, $A^*$ of Definition \[deflp\]; see [@terwgen Section 7]. In terms of parameter arrays, we have [@Terw24 Section 10]: $$\begin{aligned}
 \label{asipq0}
a_i{&\!\!\!=\!\!\!&}\theta_i+\frac{\varphi_i}{\theta^*_i-\theta^*_{i-1}}+
\frac{\varphi_{i+1}}{\theta^*_i-\theta^*_{i+1}}\\
\label{asipq1} {&\!\!\!=\!\!\!&}\theta_{d-i}+\frac{\phi_i}{\theta^*_i-\theta^*_{i-1}}+
\frac{\phi_{i+1}}{\theta^*_i-\theta^*_{i+1}},\\
\label{asipq2} a_i^*{&\!\!\!=\!\!\!&}\theta_i^*+\frac{\varphi_i}{\theta_i-\theta_{i-1}}+
\frac{\varphi_{i+1}}{\theta_i-\theta_{i+1}}\\ \label{asipq}
{&\!\!\!=\!\!\!&}\theta^*_{d-i}+\frac{\phi_{d-i+1}}{\theta_i-\theta_{i-1}}+
\frac{\phi_{d-i}}{\theta_i-\theta_{i+1}}.\end{aligned}$$ Here for $i\in\{0,d\}$ we should take $$\label{phiconv}
\varphi_0=0, \quad \varphi_{d+1}=0, \quad \phi_0=0, \quad\phi_{d+1}=0.$$ The numbers $\theta_{-1},\theta_{d+1},\theta_{-1}^*,\theta_{d+1}$ can be left undetermined. Surely, the Askey-Wilson coefficients are invariant under the action of $\downarrow$, $\Downarrow$, $\downarrow\Downarrow$ on parameter arrays.

As stated in Theorem \[lptheorem\], the coefficient sequence $\beta,\gamma,\gamma^*,\varrho,\varrho^*,\omega,\eta,\eta^*$ is unique if $d\ge 3$. If $d=2$, we can take $\beta$ freely and other coefficients get determined uniquely. If $d=1$, we can take the 3 coefficients $\beta,\gamma,\gamma^*$ freely. If $d=0$, we can take the 6 coefficients $\beta,\gamma,\gamma^*,\varrho,\varrho^*,\omega$ freely.

Normalized Askey-Wilson relations {#normawrels}
=================================

Let $(A,A^*)$ denote a Leonard pair on $V$. Suppose that it satisfies the Askey-Wilson relations $AW(\beta,\gamma,\gamma^*,\varrho,\varrho^*,\omega,\eta,\eta^*)$. It can be computed that Leonard pair (\[translation\]) then satisfies $$\begin{aligned}
 \label{newaw}
&&\hspace{-24pt}AW\left(
\beta,\;\gamma\,t+(2-\beta)\,c,\;\gamma^*t^*+(2-\beta)\,c^*,\;
\varrho\,t^2-2\gamma\,c\,t+(\beta-2)\,c^2,\right.\nonumber \\
&&\varrho^*t^*{}^2\!-2\gamma^*c^*t^*\!+(\beta\!-\!2)\,c^*{}^2,\,\omega\,t\,t^*\!
-2\gamma\,c^*t-2\gamma^*c\,t^*\!+2(\beta\!-\!2)\,c\,c^*,\nonumber\\
&&\eta\,t^2t^*-\varrho\,c^*t^2-\omega\,c\,t\,t^*
+\gamma^*c^2t^*+2\gamma^*c\,c^*t+(2-\beta)\,c^2c^*,\nonumber\\
&&\left.\eta^*t\,t^*{}^2\!-\varrho^*c\,t^*{}^2\!-\omega\,c^*t\,t^*
+\gamma\,c^*{}^2t+2\gamma\,c\,c^*t^*\!+(2-\beta)\,c\,c^*{}^2\right).\end{aligned}$$ Note 