by giving general normalized parameter arrays of each type.

By inspecting Terwilliger’s general parameter arrays [@terwgen Section 35], one can observe that the relation operators $\downarrow$, $\Downarrow$, $\downarrow\Downarrow$ do not change the Askey-Wilson type of a parameter array (but only the free parameters such as $q,h,h^*,s$ there), except that the $\Downarrow$ and $\downarrow\Downarrow$ relations mix up the quantum $q$-Krawtchouk and affine $q$-Krawtchouk types. Consequently, given a Leonard pair, all 4 associated parameter arrays have the same type, except when parameter arrays of the quantum $q$-Krawtchouk or affine $q$-Krawtchouk type occur. Therefore we can use the same classifying terminology for Leonard pairs, except that we have to merge the quantum $q$-Krawtchouk and affine $q$-Krawtchouk types.

Parameter arrays and AW relations {#paawrel}
=================================

Let us consider a parameter array as in (\[parray1\]). Suppose that the corresponding Leonard pair satisfies Askey-Wilson relations $AW(\beta,\gamma,\gamma^*,\varrho,\varrho^*,\omega,\eta,\eta^*)$. Note that the Askey-Wilson relations are invariant under isomorphism of Leonard pairs. Expressions for the 8 Askey-Wilson coefficients in terms of parameter arrays are presented in [@TerwVid Theorem 4.5 and Theorem 5.3]. Here are the formulas: $$\begin{aligned}
 \label{betap1}
\beta+1{&\!\!\!=\!\!\!&}\frac{\theta_{i-2}-\theta_{i+1}}{\theta_{i-1}-\theta_i}=
\frac{\theta^*_{i-2}-\theta^*_{i+1}}{\theta^*_{i-1}-\theta^*_i},\\
\label{betap2}\gamma{&\!\!\!=\!\!\!&}\theta_{i-1}-\beta\theta_i+\theta_{i+1},\\
\label{betap3}\gamma^*{&\!\!\!=\!\!\!&}\theta^*_{i-1}-\beta\theta^*_i+\theta^*_{i+1},\\
\label{betap4}\varrho{&\!\!\!=\!\!\!&}\theta_i^2-\beta\,\theta_i\,\theta_{i-1}
+\theta_{i-1}^2-\gamma\,(\theta_i+\theta_{i-1}),\\
\label{betap5}\varrho^*{&\!\!\!=\!\!\!&}\theta^{*\,2}_i-\beta\theta^*_i\theta^*_{i-1}
+\theta^{*2}_{i-1}- \gamma^*(\theta^*_i+\theta^*_{i-1}),\\
\label{betap6}\omega{&\!\!\!=\!\!\!&}a_i\,(\theta^*_i-\theta^*_{i+1})+a_{i-1}\,
(\theta^*_{i-1}-\theta^*_{i-2})-\gamma\,(\theta^*_i+\theta^*_{i-1})\\
\label{betap7}{&\!\!\!=\!\!\!&}a^*_i\,(\theta_i-\theta_{i+1})+a^*_{i-1}\,
(\theta_{i-1}-\theta_{i-2})-\gamma^*\,(\theta_i+\theta_{i-1}),\\
\label{betap8}\eta{&\!\!\!=\!\!\!&}a^*_i\,(\theta_{i}\!-\!\theta_{i-1})\,
(\theta_{i}\!-\!\theta_{i+1})-\gamma^*\,\theta_i^2-\omega\,\theta_i,\\
\label{betap9}\eta^*{&\!\!\!=\!\!\!&}a_i\,(\theta^*_{i}\!-\!\theta^*_{i-1})\,
(\theta^*_{i}\!-\!\theta^*_{i+1})-\gamma\,\theta_i^*{}^2-\omega\,\theta^*_i.\end{aligned}$$ The expressions for $\beta+1$ and 