by affine translations (in 4 ways, generally). The normalized coefficients $\eta,\eta^*,\omega$ in (\[orphanaw\]) are dependent on two free parameters, so there is a relation between them. Here is the relation, in the form invariant under affine rescaling: $$\begin{aligned}
 \label{orphanwc}
\left(\omega^2\!-\varrho\varrho^*\right)^2=\omega\,(\gamma\omega-\gamma^*\varrho)
(\gamma^*\omega-\gamma\varrho^*).\end{aligned}$$

Classification of AW relations {#classifics}
==============================

Askey-Wilson relations can be consistently classified by families of orthogonal polynomials in the same way as Leonard pairs. The classification is presented in the first two columns of Table \[elltab\].

  ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                     Askey-Wilson type                                                                       Askey-Wilson coefficients                                                                        with Leonard pairs
  -------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------ -----------------------------------------------------------
                         $q$-Racah                                      $\beta\neq\pm2$, $\underline{\gamma=\gamma^*\!=0}$, $\widehat{\varrho}\,\widehat{\varrho}^*\!\neq 0$                                           —

                          $q$-Hahn                              $\beta\neq\pm2$, $\underline{\gamma=\gamma^*\!=0}$, $\widehat{\varrho}=0$, $\widehat{\varrho}^*\widehat{\eta}\neq 0$                                   —

                       Dual $q$-Hahn                        $\beta\neq\pm2$, $\underline{\gamma=\gamma^*\!=0}$, $\widehat{\varrho}^*\!=0$, $\widehat{\varrho}\,\widehat{\eta}^*\!\neq 0$                               —

                       $q$-Krawtchouk                                         $\beta\neq\pm2$, $\underline{\gamma=\gamma^*\!=0}$, $\widehat{\varrho}=\widehat{\eta}=0$                            $\widehat{\varrho}^*\widehat{\eta}^*\neq 0$

                    Dual $q$-Krawtchouk                                   $\beta\neq\pm2$, $\underline{\gamma=\gamma^*\!=0}$, $\widehat{\varrho}^*\!=\widehat{\eta}^*\!=0$                         $\widehat{\varrho}\;\widehat{\eta}\neq 0$

   $\begin{array}{l} \mbox{Quantum/affine}\vspace{-5pt}\\                 $\beta\neq\pm2$, $\underline{\gamma=\gamma^*\!=0}$, $\widehat{\varrho}=\widehat{\varrho}^*\!=0$                         $\widehat{\eta}\,\widehat{\eta}^*\!\neq 0$
              \mbox{$q$-Krawtchouk}\end{array}$                                                                                                                                           

                           Racah                                                       $\beta=2$, $\gamma\,\gamma^*\neq 0$, $\underline{\varrho=\varrho^*=0}$                                                          —

                            Hahn                                                         $\beta=2$, $\gamma=0$, $\gamma^*\neq 0$, $\underline{\varrho^*=0}$                                         $\varrho\neq 0$, $\underline{\omega=0}$

                         Dual Hahn                                                      $\beta=2$, $\gamma^*\!=0$, $\gamma\!\neq 0$, $\underline{\varrho=0}$                                      $\varrho^*\!\neq 0$, $\underline{\omega=0}$

                         Krawtchouk                                                                       $\beta=2$, $\gamma=\gamma^*\!=0$                                                 $\varrho\varrho^*\!\neq 0$, $\underline{\eta=\eta^*\!=0}$

                         Bannai-Ito                                                                 $\beta=-2$, $\underline{\gamma=\gamma^*=0}$                                                 $\widehat{\varrho}\,\widehat{\varrho}^*\neq 0$
  ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

  : Classification of Askey-Wilson relations[]{data-label="elltab"}

In each line, the underlined equalities can be achieved by using affine translations if the preceding conditions are satisfied. If $\beta\neq 2$, by $\widehat{\varrho}$, $\widehat{\varrho}^*$, $\widehat{\omega}$, $\widehat{\eta}$, $\widehat{\eta}^*$ 