two sequences $$\label{scalingpars}
(\gamma,\,\varrho,\,\eta,\,\eta^*)\quad\mbox{and}\quad
(\gamma^*,\varrho^*,\eta^*,\eta).$$ By affine scalings, the two coefficients can be normalized to any convenient values. In the Askey-Wilson relations of Lemmas \[awnormals1\] and \[awnormals2\], the normalized values depend only on $\beta$: $$\begin{aligned}
 \label{scnormal}
\mbox{ }\gamma,\gamma^*: && \mbox{$2$ (if $\beta=2$)};\nonumber\\
\mbox{ }\varrho,\varrho^*: && \left\{\begin{array}{cl}
4\!-\!\beta^2, & \mbox{if } \beta\neq\pm 2,\\
1, & \mbox{if } \beta=\pm 2;\end{array}\right.\\
\mbox{ }\eta,\eta^*: &&
\left\{\begin{array}{cl}\sqrt{\beta\!+\!2}\,(\beta\!-\!2), & \mbox{if }
\eta\eta^*\neq 0 \mbox{ or } \omega=0,\\
\sqrt{\beta\!+\!2}\,(\beta\!-\!2)\,Q_{d+1}, & \mbox{if } \eta\eta^*=0 \mbox{
and } \omega\neq 0.\end{array}\right.\nonumber\end{aligned}$$ $Q_{d+1}$ can be independently defined by the linear recurrence $Q_{n+2}=\beta\,Q_n-Q_{n-2}$ with the initial values $Q_{-1}=Q_1=\sqrt{\beta\!+\!2}$, $Q_0=2$, $Q_2=\beta$. One can take for $\sqrt{\beta\!+\!2}$ any of the two values of the square root. In the context of Lemma \[awnormals1\], we should identify $\sqrt{\beta+2}$ with $q+q^{-1}$. The effect of changing the sign of $\sqrt{\beta\!+\!2}$ is multiplication of $A$ and/or $A^*$ by $-1$.

Uniqueness of normalizations {#conclusions}
============================

The results in Sections \[qpararrays1\] through \[otherpararrays\] can be used to compute the Askey-Wilson relations for any Leonard pair. To do this, one may take a parameter array corresponding to a given Leonard pair; then find an affine transformation (\[afftr\]) which normalizes the parameter array by (\[scalingtr\]) to one of the forms of Lemmas \[normlps\], 