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

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

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

-   The quantum $q$-Krawtchouk: $\theta_i=r\,q^{2i-d}$, $\theta^*_i=r\,q^{d-2i}$, $$\begin{aligned}
    \varphi_i{&\!\!\!=\!\!\!&}-\frac{q^{d+1-2i}}{r}\,\left(1-q^{2i}\right)\left(1-q^{2i-2d-2}\right),\\
    \phi_i{&\!\!\!=\!\!\!&}\frac{q^{2d+2-4i}}{r}\,
    \left(1-q^{2i}\right)\left(1-q^{2i-2d-2}\right)\left(r^3-q^{2i-d-1}\right).\end{aligned}$$

-   The affine $q$-Krawtchouk: $\theta_i=r\,q^{d-2i}$, $\theta^*_i=r\,q^{d-2i}$, $$\begin{aligned}
    \varphi_i{&\!\!\!=\!\!\!&}\frac{q^{2d+2-4i}}r\,\left(1-q^{2i}\right)\left(1-q^{2i-2d-2}\right)\left(r^3-q^{2i-d-1}\right),\\
    \phi_i{&\!\!\!=\!\!\!&}-\frac{q^{d+1-2i}}r\,\left(1-q^{2i}\right)\left(1-q^{2i-2d-2}\right).\end{aligned}$$

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.** ]{}]{}In every case of Lemma \[normlps\], we substitute $$q\mapsto q^2,\qquad s\mapsto\frac1{s^2\,q^{2d+2}},\qquad
s^*\mapsto\frac1{s^*{}^2q^{2d+2}}.$$ Besides, in the $q$-Racah, $q$-Hahn, dual $q$-Hahn, quantum $q$-Krawtchouk and affine $q$-Krawtchouk cases we substitute $r$ by, respectively, $$\frac{r}{s\,s^*q^{d+1}},\quad \frac{1}{s^*r^2q^{d+1}},\quad
\frac{1}{s\,r^2q^{d+1}},\quad \frac{q^{d+1}}{r^3},\quad
\frac{1}{r^3q^{d+1}}.$$ After that, we apply affine scaling. We use formula (\[scalingtr\]) with $c=0,c^*=0$ and $(t,t^*)$ equal to, respectively in the listed order, $$\begin{aligned}
(s\,q^d,s^*q^d), \quad (r\,q^d,s^*q^d), \quad (s\,q^d,r\,q^d), \quad
(q^d,s^*q^d), \quad (s\,q^d,q^d),\\
(r\,q^{-d-2},r\,q^d), \quad (r\,q^d,r\,q^d).\end{aligned}$$ [$\Box$]{}

\[awnormals1\] As in the previous lemma, let $q,s,s^*,r$ denote nonzero scalar parameters. We use the following notations: $$\begin{aligned}
\label{normconsts1}
Q_j=q^j+q^{-j},\qquad Q_j^*=q^j-q^{-j}, \quad \mbox{for $j=1,2,\ldots$},\\
S=s+\frac{1}s, \qquad S^*=s^*+\frac{1}{s^*}, \qquad R=r+\frac{1}{r}.\end{aligned}$$ The Askey-Wilson relations for the parameter arrays of Lemma $\ref{normlps1}$ are:

-   For the $q$-Racah case: $$\begin{aligned}
    &&\hspace{-24pt}AW\big(Q_2,\,0,\,0,\,-Q_2^*{}^2,\,-Q_2^*{}^2,
    \,-Q_1^*{}^2\!\left(S\,S^*\!+Q_{d+1}R\right),\nonumber\\
    &&Q_1Q_1^*{}^2\!\left( S\,R+Q_{d+1}S^*\right),\,
    Q_1Q_1^*{}^2\!\left(S^*R+Q_{d+1}S\right)\big).\end{aligned}$$

-   For the $q$-Hahn case: 