\varphi_i{&\!\!\!=\!\!\!&}v\,i\,(i-d-1),\\
    \phi_i{&\!\!\!=\!\!\!&}(v-1)\,i\,(i-d-1).\end{aligned}$$

-   The Bannai-Ito case: $\theta_i=(-1)^i\left(i+u-\frac{d}2\right)$, $\theta^*_i=(-1)^i\left(i+u^*\!-\frac{d}2\right)$, $$\begin{aligned}
    \varphi_i=\left\{ \begin{array}{cl} -i\left(i+u+u^*\!+v-\frac{d+1}2\right),&
    \mbox{for $i$ even, $d$ even}.\\
    -(i-d-1)\left(i+u+u^*\!-v-\frac{d+1}2\right), &
    \mbox{for $i$ odd, $d$ even}.\\
    -i\,(i-d-1), & \mbox{for $i$ even, $d$ odd}.\\
    v^2-\left(i+u+u^*\!-\frac{d+1}2\right)^2, &
     \mbox{for $i$ odd, $d$ odd}.\end{array} \right.\end{aligned}$$ $$\begin{aligned}
    \phi_i=\left\{ \begin{array}{cl} i\left(i-u+u^*\!-v-\frac{d+1}2\right),& \mbox{for $i$ even, $d$ even}.\\
    (i-d-1)\left(i-u+u^*\!+v-\frac{d+1}2\right), & \mbox{for $i$ odd, $d$ even}.\\
    -i\,(i-d-1), & \mbox{for $i$ even, $d$ odd}.\\
    v^2-\left(i-u+u^*\!-\frac{d+1}2\right)^2, &
     \mbox{for $i$ odd, $d$ odd}.\end{array} \right.\end{aligned}$$

In each case, $u,u^*,v$ are 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.** ]{}]{}Like in the proof of Lemma \[normlps\], we adjust Terwilliger’s parameters $\theta_0,\theta^*_0$ by an affine translation, and then adjust other two parameters by an affine scaling. We also make linear substitutions for the remaining parameters. In the Racah case, we substitute $$s\mapsto 2u,\quad s^*\!\mapsto 2u^*,\quad r_1\mapsto u+u^*-v, \quad \mbox{so
that} \quad r_2=u+u^*+d+1+v.$$ Then we adjust $\theta_0=u^2+u$, $\theta^*_0=u^*{}^2\!+u^*\!$, $h=1$, $h^*\!=1$. In the Hahn case, we substitute $s^*\!\mapsto 2u^*$, $r\mapsto
u^*\!+2v$ and adjust $\theta_0=v-\frac{d}2$, $\theta^*_0=u^*{}^2\!+u^*\!$, $h^*\!=1$, $s=1$. In the dual Hahn case, we substitute $s\mapsto 2u$, $r\mapsto u+2v$ and adjust $\theta_0=u^2+u$, $\theta^*_0=v-\frac{d}2$, $h=1$, $s=1$. In the Krawtchouk case, we substitute $r\mapsto v$ and adjust $\theta_0=-\frac{d}2$, 