-   For the dual $q$-Hahn case: $$\begin{aligned}
    &&\hspace{-24pt}AW\big(q+q^{-1},\,0,\,0,\,s\,K,\,0,\,-K^*\!\left(S+r\,Q\right),\nonumber\\
    &&\qquad (q\!+\!1)K^*\!\left(S\,r+s\,Q\right),\,(q\!+\!1)K^*r\big).\end{aligned}$$

-   For the $q$-Krawtchouk case: $$\begin{aligned}
    AW\big(q+q^{-1},\,0,\,0,\,0,\,s^*K,\,-K^*\!S^*,\,0,\,(q\!+\!1)K^*s^*Q\big).\end{aligned}$$

-   For the dual $q$-Krawtchouk case: $$\begin{aligned}
    AW\big(q+q^{-1},\,0,\,0,\,s\,K,\,0,\,-K^*\!S,\,(q\!+\!1)K^*s\,Q,\,0\big).\end{aligned}$$

-   For the quantum $q$-Krawtchouk case: $$\begin{aligned}
    AW\big(q+q^{-1},\,0,\,0,\,0,\,0,\,-K^*\!\left(q^{d+1}\!+r\,Q\right),\,
    (q\!+\!1)(q\!-\!1)^2r,\,(q\!+\!1)K^*r\big).$$

-   For the affine $q$-Krawtchouk case: $$\begin{aligned}
    AW\big(q+q^{-1},\,0,\,0,\,0,\,0,\,-K^*\!\left(1+r\,Q\right),\,
    (q\!+\!1)K^*r,\,(q\!+\!1)K^*r\big).\quad\end{aligned}$$

[[**Proof.** ]{}]{}Direct computations with formulas (\[betap1\])–(\[asipq2\]).[$\Box$]{}

Alternative normalized $q$-arrays {#qpararrays2}
=================================

Here we present alternative normalizations of the general parameter arrays in [@terwgen Section 35] with the general $q$-parameter. The parameters are rescaled, and the free parameters $q,s,s^*,r$ are different. In particular, the $q$ of the previous section is replaced by $q^2$. The normalization for the $q$-Racah case is proposed in [@hjalmarpaul]. The corresponding Askey-Wilson relations are normalized according to part 1 of Lemma \[normrules\]. Advantages of this normalization are: the two nonzero values normalized by affine scaling are $q$-constants; expressions for normalized parameter arrays are more symmetric; the set of normalized parameter arrays is preserved by the $\downarrow$, $\Downarrow$, $\downarrow\Downarrow$ operations (see Section \[conclusions\]).

\[normlps1\] The parameter arrays in [[@terwgen Examples 35.2–35.8]]{} can be normalized by affine transformations $(\ref{scalingtr})$ to the following forms:

-   The $q$-Racah case: $\displaystyle\theta_i=s\,q^{d-2i}+\frac{q^{2i-d}}{s},\quad
    \theta^*_i=s^*q^{d-2i}+\frac{q^{2i-d}}{s^*}$. $$\begin{aligned}
    \varphi_i{&\!\!\!=\!\!\!&}\frac{q^{2d+2-4i}}{s\,s^*r}\left(1-q^{2i}\right)\left(1-q^{2i-2d-2}\right)
    \left(s\,s^*-r\,q^{2i-d-1}\right)\left(s\,s^*r-q^{2i-d-1}\right),\\
    \phi_i{&\!\!\!=\!\!\!&}\frac{q^{2d+2-4i}}{s\,s^*r}\left(1-q^{2i}\right)\nonumber
    \left(1-q^{2i-2d-2}\right)\left(s^*r-s\,q^{2i-d-1}\right)\left(s^*-s\,r\,q^{2i-d-1}\right).\end{aligned}$$

-   The $q$-Hahn case: $\displaystyle\theta_i=r\,q^{d-2i},\quad
    \theta^*_i=s^*q^{d-2i}+\frac{q^{2i-d}}{s^*}$, $$\begin{aligned}
    \varphi_i{&\!\!\!=\!\!\!&}\frac{q^{2d+2-4i}}{r}\,
    \left(1-q^{2i}\right)\left(1-q^{2i-2d-2}\right)\left(s^*r^2-q^{2i-d-1}\right),\\
    \phi_i{&\!\!\!=\!\!\!&}-\frac{q^{d+1-2i}}{r\,s^*}\,
    \left(1-q^{2i}\right)\left(1-q^{2i-2d-2}\right)\left(s^*-r^2q^{2i-d-1}\right).\end{aligned}$$

-   