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

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

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

-   The affine $q$-Krawtchouk case: $\theta_i=q^{-i}$, $\theta^*_i=q^{-i}$, $$\begin{aligned}
    \varphi_i{&\!\!\!=\!\!\!&}q^{1-2i}\,\left(1-q^i\right)\left(1-q^{i-d-1}\right)\left(1-r\,q^i\right),\\
    \phi_i{&\!\!\!=\!\!\!&}-r\,q^{1-i}\,\left(1-q^i\right)\left(1-q^{i-d-1}\right).\end{aligned}$$

In each case, $q,s,s^*,r$ are nonzero scalar parameters such that $\theta_i\neq \theta_j$, $\theta^*_i\neq\theta^*_j$ for $0\le i<j\le d$, and $\varphi_i\neq 0$, $\phi_i\neq 0$ for $1\le i \le d$.

[[**Proof.** ]{}]{}By affine translations, we adjust Terwilliger’s parameters $\theta_0,\theta^*_0$ so that we have only summands depending on $i$ in the expanded expressions for $\theta_i$, $\theta^*_i$ in [@terwgen Examples 35.2–35.8]. By affine scalings, we set Terwilliger’s parameters $h,h^*$ to the value 1. In the quantum $q$-Krawtchouk case [@terwgen Example 35.5] there is no parameter $h$, so we set $s=1$. Other parameters remain unchanged, except that in the $q$-Racah case we rename $r_1$ to $r$ and set $r_2=s\,s^*q^{d+1}/r$.[$\Box$]{}

\[awnormals\] Let $q,s,s^*,r$ denote the same scalar parameters as in the previous lemma. We use the following notations: $$\begin{aligned}
\label{normconsts}
S=s\,q^{d+1}+1, \qquad S^*=s^*q^{d+1}+1, \qquad
R=r+\frac{s\,s^*\,q^{d+1}}r,\\ Q=q^{d+1}+1,\qquad K=-\frac{(q^2\!-1)^2}q, \quad
K^*=\frac{(q-1)^2}{q^{d+1}}.\end{aligned}$$ The Askey-Wilson relations for the parameter arrays of Lemma $\ref{normlps}$ are:

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

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

