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## 4 matching pages

##### 1: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes
For Askey–Wilson $p_{n}\left(\cos\theta;a,b,c,d\,|\,q\right)$ the leading term is given by …
18.28.1_5 $R_{n}(z)=R_{n}(z;a,b,c,d\,|\,q)=\frac{p_{n}\left(\frac{1}{2}(z+z^{-1});a,b,c,d% \,|\,q\right)}{a^{-n}\left(ab,ac,ad;q\right)_{n}}={{}_{4}\phi_{3}}\left({q^{-n% },abcdq^{n-1},az,az^{-1}\atop ab,ac,ad};q,q\right).$
The polynomials $p_{n}\left(x;a,b,c,d\,|\,q\right)$ are symmetric in the parameters $a,b,c,d$. …
18.28.7 $Q_{n}\left(\cos\theta;a,b\,|\,q\right)=p_{n}\left(\cos\theta;a,b,0,0\,|\,q% \right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}\frac{\left(abq^{\ell};q\right)_{n-\ell% }\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}\*\prod_{j=0}^{\ell-1}(% 1-2aq^{j}\cos\theta+a^{2}q^{2j})=\frac{\left(ab;q\right)_{n}}{a^{n}}{{}_{3}% \phi_{2}}\left({q^{-n},a{\mathrm{e}}^{\mathrm{i}\theta},a{\mathrm{e}}^{-% \mathrm{i}\theta}\atop ab,0};q,q\right)=\left(b{\mathrm{e}}^{-\mathrm{i}\theta% };q\right)_{n}{\mathrm{e}}^{\mathrm{i}n\theta}{{}_{2}\phi_{1}}\left({q^{-n},a{% \mathrm{e}}^{\mathrm{i}\theta}\atop b^{-1}q^{1-n}{\mathrm{e}}^{\mathrm{i}% \theta}};q,b^{-1}q{\mathrm{e}}^{-\mathrm{i}\theta}\right).$
18.28.29 $\lim_{q\to 1}\frac{p_{n}\left(1-\tfrac{1}{2}x(1-q)^{2};q^{a},q^{b},q^{c},q^{d}% \,|\,q\right)}{(1-q)^{3n}}=W_{n}\left(x;a,b,c,d\right).$
##### 3: 18.1 Notation
• Askey–Wilson: $p_{n}\left(x;a,b,c,d\,|\,q\right)$.

• ##### 4: Errata
• Equation (18.28.1)
18.28.1 $p_{n}(x)=p_{n}\left(x;a,b,c,d\,|\,q\right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}% \left(abq^{\ell},acq^{\ell},adq^{\ell};q\right)_{n-\ell}\*\frac{\left(q^{-n},% abcdq^{n-1};q\right)_{\ell}}{\left(q;q\right)_{\ell}}\prod_{j=0}^{\ell-1}{(1-2% aq^{j}x+a^{2}q^{2j})},$
18.28.1_5 $R_{n}(z)=R_{n}(z;a,b,c,d\,|\,q)=\frac{p_{n}\left(\frac{1}{2}(z+z^{-1});a,b,c,d% \,|\,q\right)}{a^{-n}\left(ab,ac,ad;q\right)_{n}}={{}_{4}\phi_{3}}\left({q^{-n% },abcdq^{n-1},az,az^{-1}\atop ab,ac,ad};q,q\right)$

Previously we presented all the information of these formulas in one equation

$p_{n}(\cos\theta)=p_{n}\left(\cos\theta;a,b,c,d\,|\,q\right)=a^{-n}\sum_{\ell=% 0}^{n}q^{\ell}\left(abq^{\ell},acq^{\ell},adq^{\ell};q\right)_{n-\ell}\*\frac{% \left(q^{-n},abcdq^{n-1};q\right)_{\ell}}{\left(q;q\right)_{\ell}}\prod_{j=0}^% {\ell-1}{(1-2aq^{j}\cos\theta+a^{2}q^{2j})}=a^{-n}\left(ab,ac,ad;q\right)_{n}% \*{{}_{4}\phi_{3}}\left({q^{-n},abcdq^{n-1},a{\mathrm{e}}^{\mathrm{i}\theta},a% {\mathrm{e}}^{-\mathrm{i}\theta}\atop ab,ac,ad};q,q\right).$