§18.28(i) Introduction

The Askey–Wilson class OP’s comprise the four-parameter families of Askey–Wilson polynomials and of $q$-Racah polynomials, and cases of these families obtained by specialization of parameters. The Askey–Wilson polynomials form a system of OP’s $\{p_{n}(x)\}$, $n=0,1,2,\dots$, that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. The $q$-Racah polynomials form a system of OP’s $\{p_{n}(x)\}$, $n=0,1,2,\dots,N$, that are orthogonal with respect to a weight function on a sequence $\{q^{-y}+cq^{y+1}\}$, $y=0,1,\dots,N$, with $c$ a constant. Both the Askey–Wilson polynomials and the $q$-Racah polynomials can best be described as functions of $z$ (resp. $y$) such that $P_{n}(z)=p_{n}(\tfrac{1}{2}(z+z^{-1}))$ in the Askey–Wilson case, and $P_{n}(y)=p_{n}(q^{-y}+cq^{y+1})$ in the $q$-Racah case, and both are eigenfunctions of a second order $q$-difference operator similar to (18.27.1).

In the remainder of this section the Askey–Wilson class OP’s are defined by their $q$-hypergeometric representations, followed by their orthogonal properties. For further properties see Koekoek et al. (2010, Chapter 14). See also Gasper and Rahman (2004, pp. 180–199), Ismail (2009, Chapter 15), and Koornwinder (2012). For the notation of $q$-hypergeometric functions see §§17.2 and 17.4(i).

 18.28.1 $p_{n}(x)=p_{n}\left(x;a,b,c,d\,|\,q\right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}% \left(abq^{\ell},acq^{\ell},adq^{\ell};q\right)_{n-\ell}\*\frac{\left(q^{-n},% abcdq^{n-1};q\right)_{\ell}}{\left(q;q\right)_{\ell}}\prod_{j=0}^{\ell-1}{(1-2% aq^{j}x+a^{2}q^{2j})},$ ⓘ Defines: $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\,|\,\NVar{q}\right)$: Askey–Wilson polynomial Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: multiple $q$-Pochhammer symbol, $p_{n}(x)$: polynomial of degree $n$, $q$: real variable, $\ell$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: (18.28.29), §18.28(ii), §18.38(iii), Erratum (V1.2.0) for Equation (18.28.1) Permalink: http://dlmf.nist.gov/18.28.E1 Encodings: TeX, pMML, png See also: Annotations for §18.28(ii), §18.28 and Ch.18
 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 $a,b,c,d$.

Orthogonality

Assume $a,b,c,d$ are all real, or two of them are real and two form a conjugate pair, or none of them are real but they form two conjugate pairs.

 18.28.2 $\int_{-1}^{1}p_{n}(x)p_{m}(x)w(x)\,\mathrm{d}x=h_{n}\delta_{n,m},$ $|a|,|b|,|c|,|d|\leq 1$, $ab,ac,ad,bc,bd,cd\neq 1$, ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$: Kronecker delta, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $p_{n}(x)$: polynomial of degree $n$, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $h_{n}$ Referenced by: Erratum (V1.2.0) for Equation (18.28.2) Permalink: http://dlmf.nist.gov/18.28.E2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This constraint of this equation was updated to include $ab,ac,ad,bc,bd,cd\neq 1$. See also: Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

where

 18.28.3 $2\pi\sin\theta\,w(\cos\theta)={\left|\frac{\left({\mathrm{e}}^{2i\theta};q% \right)_{\infty}}{\left(a{\mathrm{e}}^{\mathrm{i}\theta},b{\mathrm{e}}^{% \mathrm{i}\theta},c{\mathrm{e}}^{\mathrm{i}\theta},d{\mathrm{e}}^{\mathrm{i}% \theta};q\right)_{\infty}}\right|}^{2},$
 18.28.4 $h_{0}=\frac{\left(abcd;q\right)_{\infty}}{\left(q,ab,ac,ad,bc,bd,cd;q\right)_{% \infty}},$
 18.28.5 $h_{n}=h_{0}\frac{(1-abcdq^{n-1})\left(q,ab,ac,ad,bc,bd,cd;q\right)_{n}}{(1-% abcdq^{2n-1})\left(abcd;q\right)_{n}},$ $n=1,2,\dotsc$.

More generally,

 18.28.6 $\int_{-1}^{1}p_{n}(x)p_{m}(x)w(x)\,\mathrm{d}x+\sum_{\ell}p_{n}(x_{\ell})p_{m}% (x_{\ell})\omega_{\ell}=h_{n}\delta_{n,m},$ $ab,ac,ad,bc,bd,cd\in\{z\in\mathbb{C}\mid|z|\leq 1,\,z\neq 1\}$, ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$: Kronecker delta, $\mathbb{C}$: complex plane, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\in$: element of, $\int$: integral, $p_{n}(x)$: polynomial of degree $n$, $w(x)$: weight function, $z$: complex variable, $\ell$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $h_{n}$ Referenced by: Erratum (V1.2.0) for Equation (18.28.6) Permalink: http://dlmf.nist.gov/18.28.E6 Encodings: TeX, pMML, png Modification (effective with 1.2.0): This constraint of this equation was updated to include $ab,ac,ad,bc,bd,cd\in\{z\in\mathbb{C}\mid|z|\leq 1,\,z\neq 1\}$. See also: Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

with $w(x)$ and $h_{n}$ as above. Also, $x_{\ell}$ are the points $\tfrac{1}{2}(\alpha q^{\ell}+\alpha^{-1}q^{-\ell})$ with $\alpha$ any of the $a,b,c,d$ whose absolute value exceeds $1$, and the sum is over the $\ell=0,1,2,\dots$ with $|\alpha q^{\ell}|>1$. See Koekoek et al. (2010, Eq. (14.1.3)) for the value of $\omega_{\ell}$ when $\alpha=a$.

$q$-Difference Equation

 18.28.6_1 $(LR_{n})(z)=(q^{-n}+abcdq^{n-1})R_{n}(z),$ ⓘ Symbols: $z$: complex variable, $q$: real variable and $n$: nonnegative integer Referenced by: §18.28(ii), §18.28(ii), Erratum (V1.2.0) §18.28 Permalink: http://dlmf.nist.gov/18.28.E6_1 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

where the operator $L$ is defined by

 18.28.6_2 $(Lf)(z)=\frac{(1-az)(1-bz)(1-cz)(1-dz)}{(1-z^{2})(1-qz^{2})}\*\bigl{(}f(qz)-f(% z)\bigr{)}+\frac{(1-az^{-1})(1-bz^{-1})(1-cz^{-1})(1-dz^{-1})}{(1-z^{-2})(1-qz% ^{-2})}\*\bigl{(}f(q^{-1}z)-f(z)\bigr{)}+\big{(}1+q^{-1}abcd\big{)}f(z).$ ⓘ Symbols: $z$: complex variable and $q$: real variable Referenced by: §18.28(ii), §18.38(iii), §18.38(iii) Permalink: http://dlmf.nist.gov/18.28.E6_2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

Recurrence Relation

 18.28.6_3 $(z+z^{-1})R_{n}(z)=a_{n}\big{(}R_{n+1}(z)-R_{n}(z)\big{)}+c_{n}\*\big{(}R_{n-1% }(z)-R_{n}(z)\big{)}+(a+a^{-1})R_{n}(z),$ ⓘ Symbols: $z$: complex variable and $n$: nonnegative integer Referenced by: §18.28(ii) Permalink: http://dlmf.nist.gov/18.28.E6_3 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

where $c_{0}=0$ and

 18.28.6_4 $\displaystyle a_{n}$ $\displaystyle=\frac{(1-abq^{n})(1-acq^{n})(1-adq^{n})(1-abcdq^{n-1})}{a(1-% abcdq^{2n-1})(1-abcdq^{2n})},$ $\displaystyle c_{n}$ $\displaystyle=\frac{a(1-q^{n})(1-bcq^{n-1})(1-bdq^{n-1})(1-cdq^{n-1})}{(1-% abcdq^{2n-2})(1-abcdq^{2n-1})}.$ ⓘ Symbols: $q$: real variable and $n$: nonnegative integer Referenced by: §18.28(ii) Permalink: http://dlmf.nist.gov/18.28.E6_4 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

Duality

Define dual parameters $\tilde{a},\tilde{b},\tilde{c},\tilde{d}$ in terms of $a,b,c,d$ by

 $\tilde{a}=(q^{-1}abcd)^{\frac{1}{2}},\quad\tilde{b}=\ifrac{ab}{\tilde{a}},% \quad\tilde{c}=\ifrac{ac}{\tilde{a}},\quad\tilde{d}=\ifrac{ad}{\tilde{a}},$

assuming $a,b,c,d\neq 0$. Then

 18.28.6_5 $R_{n}(a^{-1}q^{-m};a,b,c,d\,|\,q)=R_{m}(\tilde{a}^{-1}q^{-n};\tilde{a},\tilde{% b},\tilde{c},\tilde{d}\,|\,q),$ $m,n=0,1,2,\ldots$. ⓘ Symbols: $q$: real variable, $m$: nonnegative integer and $n$: nonnegative integer Referenced by: §18.28(ii), §18.28(ii), Erratum (V1.2.0) §18.28 Permalink: http://dlmf.nist.gov/18.28.E6_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

§18.28(iii) Al-Salam–Chihara Polynomials

 18.28.7 $Q_{n}\left(\cos\theta;a,b\,|\,q\right)=p_{n}\left(\cos\theta;a,b,0,0\,|\,q% \right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}\frac{\left(abq^{\ell};q\right)_{n-\ell% }\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}\*\prod_{j=0}^{\ell-1}(% 1-2aq^{j}\cos\theta+a^{2}q^{2j})=\frac{\left(ab;q\right)_{n}}{a^{n}}{{}_{3}% \phi_{2}}\left({q^{-n},a{\mathrm{e}}^{\mathrm{i}\theta},a{\mathrm{e}}^{-% \mathrm{i}\theta}\atop ab,0};q,q\right)=\left(b{\mathrm{e}}^{-\mathrm{i}\theta% };q\right)_{n}{\mathrm{e}}^{\mathrm{i}n\theta}{{}_{2}\phi_{1}}\left({q^{-n},a{% \mathrm{e}}^{\mathrm{i}\theta}\atop b^{-1}q^{1-n}{\mathrm{e}}^{\mathrm{i}% \theta}};q,b^{-1}q{\mathrm{e}}^{-\mathrm{i}\theta}\right).$
 18.28.8 $\frac{1}{2\pi}\int_{0}^{\pi}Q_{n}\left(\cos\theta;a,b\,|\,q\right)Q_{m}\left(% \cos\theta;a,b\,|\,q\right)\*{\left|\frac{\left({\mathrm{e}}^{2i\theta};q% \right)_{\infty}}{\left(a{\mathrm{e}}^{\mathrm{i}\theta},b{\mathrm{e}}^{% \mathrm{i}\theta};q\right)_{\infty}}\right|}^{2}\,\mathrm{d}\theta=\frac{% \delta_{n,m}}{\left(q^{n+1},abq^{n};q\right)_{\infty}},$ $a,b\in\mathbb{R}$ or $a=\overline{b}$; $ab\neq 1$; $|a|,|b|\leq 1$.

More generally, if $|ab|\leq 1$ instead of $|a|,|b|\leq 1$, discrete terms need to be added to the right-hand side of (18.28.8); see Koekoek et al. (2010, Eq. (14.8.3)).

§18.28(iv) $q^{-1}$-Al-Salam–Chihara Polynomials

 18.28.9 $Q_{n}\left(\tfrac{1}{2}(aq^{-y}+a^{-1}q^{y});a,b\,|\,q^{-1}\right)=(-1)^{n}b^{% n}q^{-\frac{1}{2}n(n-1)}\*\left((ab)^{-1};q\right)_{n}{{}_{3}\phi_{1}}\left({q% ^{-n},q^{-y},a^{-2}q^{y}\atop(ab)^{-1}};q,q^{n}ab^{-1}\right).$ ⓘ Defines: $Q_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\,|\,\NVar{q}^{-1}\right)$: $q^{-1}$-Al-Salam–Chihara polynomial Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$: basic hypergeometric (or $q$-hypergeometric) function, $y$: real variable, $q$: real variable, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.28.E9 Encodings: TeX, pMML, png See also: Annotations for §18.28(iv), §18.28 and Ch.18
 18.28.10 $\sum_{y=0}^{\infty}\frac{(1-q^{2y}a^{-2})\left(a^{-2},(ab)^{-1};q\right)_{y}}{% (1-a^{-2})\left(q,bqa^{-1};q\right)_{y}}(ba^{-1})^{y}q^{y^{2}}\*Q_{n}\left(% \tfrac{1}{2}(aq^{-y}+a^{-1}q^{y});a,b\,|\,q^{-1}\right)\*Q_{m}\left(\tfrac{1}{% 2}(aq^{-y}+a^{-1}q^{y});a,b\,|\,q^{-1}\right)=\frac{\left(qa^{-2};q\right)_{% \infty}}{\left(ba^{-1}q;q\right)_{\infty}}\left(q,(ab)^{-1};q\right)_{n}(ab)^{% n}q^{-n^{2}}\delta_{n,m}.$

Eq. (18.28.10) is valid when either

 18.28.11 $01,a^{-1}b ⓘ Symbols: $\in$: element of, $\mathbb{R}$: real line and $q$: real variable Referenced by: §18.28(iv) Permalink: http://dlmf.nist.gov/18.28.E11 Encodings: TeX, pMML, png See also: Annotations for §18.28(iv), §18.28 and Ch.18

or

 18.28.12 $0% 0,a^{-1}b ⓘ Symbols: $\in$: element of, $\Im$: imaginary part, $\mathrm{i}$: imaginary unit, $\mathbb{R}$: real line and $q$: real variable Referenced by: §18.28(iv) Permalink: http://dlmf.nist.gov/18.28.E12 Encodings: TeX, pMML, png See also: Annotations for §18.28(iv), §18.28 and Ch.18

If, in addition to (18.28.11) or (18.28.12), we have $a^{-1}b\leq q$, then the measure in (18.28.10) is the unique orthogonality measure. Also, if $q, then (18.28.10) holds with $a,b$ interchanged. For further nondegenerate cases see Chihara and Ismail (1993) and Christiansen and Ismail (2006).

§18.28(v) Continuous $q$-Ultraspherical Polynomials

 18.28.13 $\displaystyle C_{n}\left(\cos\theta;\beta\,|\,q\right)$ $\displaystyle=\sum_{\ell=0}^{n}\frac{\left(\beta;q\right)_{\ell}\left(\beta;q% \right)_{n-\ell}}{\left(q;q\right)_{\ell}\left(q;q\right)_{n-\ell}}{\mathrm{e}% }^{\mathrm{i}(n-2\ell)\theta}=\frac{\left(\beta;q\right)_{n}}{\left(q;q\right)% _{n}}{\mathrm{e}}^{\mathrm{i}n\theta}{{}_{2}\phi_{1}}\left({q^{-n},\beta\atop% \beta^{-1}q^{1-n}};q,\beta^{-1}q{\mathrm{e}}^{-2\mathrm{i}\theta}\right).$ ⓘ Defines: $C_{\NVar{n}}\left(\NVar{x};\NVar{\beta}\,|\,\NVar{q}\right)$: continuous $q$-ultraspherical polynomial Symbols: $\cos\NVar{z}$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: real variable, $\ell$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.28.E13 Encodings: TeX, pMML, png See also: Annotations for §18.28(v), §18.28 and Ch.18 18.28.14 $\displaystyle C_{n}\left(\cos\theta;\beta\,|\,q\right)$ $\displaystyle=\frac{\left(\beta^{2};q\right)_{n}}{\left(q;q\right)_{n}\beta^{% \frac{1}{2}n}}{{}_{4}\phi_{3}}\left({q^{-n},\beta^{2}q^{n},\beta^{\frac{1}{2}}% {\mathrm{e}}^{\mathrm{i}\theta},\beta^{\frac{1}{2}}{\mathrm{e}}^{-\mathrm{i}% \theta}\atop\beta q^{\frac{1}{2}},-\beta,-\beta q^{\frac{1}{2}}};q,q\right).$
 18.28.15 $\frac{1}{2\pi}\int_{0}^{\pi}C_{n}\left(\cos\theta;\beta\,|\,q\right)C_{m}\left% (\cos\theta;\beta\,|\,q\right)\*{\left|\frac{\left({\mathrm{e}}^{2\mathrm{i}% \theta};q\right)_{\infty}}{\left(\beta{\mathrm{e}}^{2\mathrm{i}\theta};q\right% )_{\infty}}\right|}^{2}\,\mathrm{d}\theta=\frac{\left(\beta,\beta q;q\right)_{% \infty}}{\left(\beta^{2},q;q\right)_{\infty}}\frac{(1-\beta)\left(\beta^{2};q% \right)_{n}}{(1-\beta q^{n})\left(q;q\right)_{n}}\delta_{n,m},$ $-1<\beta<1$.

These polynomials are also called Rogers polynomials.

§18.28(vi) Continuous $q$-Hermite Polynomials

 18.28.16 $H_{n}\left(\cos\theta\,|\,q\right)=\sum_{\ell=0}^{n}\frac{\left(q;q\right)_{n}% {\mathrm{e}}^{\mathrm{i}(n-2\ell)\theta}}{\left(q;q\right)_{\ell}\left(q;q% \right)_{n-\ell}}={\mathrm{e}}^{\mathrm{i}n\theta}{{}_{2}\phi_{0}}\left({q^{-n% },0\atop-};q,q^{n}{\mathrm{e}}^{-2\mathrm{i}\theta}\right).$
 18.28.17 $\frac{1}{2\pi}\int_{0}^{\pi}H_{n}\left(\cos\theta\,|\,q\right)H_{m}\left(\cos% \theta\,|\,q\right){\left|\left({\mathrm{e}}^{2\mathrm{i}\theta};q\right)_{% \infty}\right|}^{2}\,\mathrm{d}\theta=\frac{\delta_{n,m}}{\left(q^{n+1};q% \right)_{\infty}}.$

§18.28(vii) Continuous $q^{-1}$-Hermite Polynomials

 18.28.18 $h_{n}\left(\sinh t\,|\,q\right)=\sum_{\ell=0}^{n}q^{\frac{1}{2}\ell(\ell+1)}% \frac{\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}{\mathrm{e}}^{(n-2% \ell)t}={\mathrm{e}}^{nt}{{}_{1}\phi_{1}}\left({q^{-n}\atop 0};q,-q{\mathrm{e}% }^{-2t}\right)={\mathrm{i}}^{-n}H_{n}\left(\mathrm{i}\sinh t\,|\,q^{-1}\right).$

For continuous $q^{-1}$-Hermite polynomials the orthogonality measure is not unique. See Askey (1989) and Ismail and Masson (1994) for examples.

§18.28(viii) $q$-Racah Polynomials

With $x=q^{-y}+\gamma\delta q^{y+1}$,

 18.28.19 $R_{n}(x)=R_{n}\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)=\sum_{\ell=0}^{n% }\frac{q^{\ell}\left(q^{-n},\alpha\beta q^{n+1};q\right)_{\ell}}{\left(\alpha q% ,\beta\delta q,\gamma q,q;q\right)_{\ell}}\*\prod_{j=0}^{\ell-1}(1-q^{j}x+% \gamma\delta q^{2j+1})={{}_{4}\phi_{3}}\left({q^{-n},\alpha\beta q^{n+1},q^{-y% },\gamma\delta q^{y+1}\atop\alpha q,\beta\delta q,\gamma q};q,q\right),$ $\alpha q$, $\beta\delta q$, or $\gamma q=q^{-N}$; $n=0,1,\dots,N$. ⓘ Defines: $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\,|\,\NVar{q}\right)$: $q$-Racah polynomial and $R_{n}(x)$ (locally) Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$: basic hypergeometric (or $q$-hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: multiple $q$-Pochhammer symbol, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $q$: real variable, $\ell$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.28.E19 Encodings: TeX, pMML, png See also: Annotations for §18.28(viii), §18.28 and Ch.18
 18.28.20 $\sum_{y=0}^{N}R_{n}(q^{-y}+\gamma\delta q^{y+1})R_{m}(q^{-y}+\gamma\delta q^{y% +1})\omega_{y}=h_{n}\delta_{n,m},$ $n,m=0,1,\dots,N$,

with

 18.28.21 $\omega_{y}=\frac{\left(\alpha q,\beta\delta q,\gamma q,\gamma\delta q;q\right)% _{y}}{\left(q,\frac{\gamma\delta}{\alpha}q,\frac{\gamma}{\beta}q,\delta q;q% \right)_{y}}\frac{1-\gamma\delta q^{2y+1}}{\left(\alpha\beta q\right)^{y}},$
 18.28.22 $h_{n}=\frac{\left(\alpha\beta\right)^{n+1}q^{\left(n+1\right)^{2}}}{\alpha% \beta q^{2n+1}-1}\frac{\left(q;q\right)_{n}}{\left(\alpha q,\beta\delta q,% \gamma q;q\right)_{n}}\*\frac{\left(\frac{\gamma}{\alpha\beta}q^{-n},\frac{% \delta}{\alpha}q^{-n},\frac{1}{\beta}q^{-n},\gamma\delta q;q\right)_{\infty}}{% \left(\frac{1}{\alpha\beta}q^{-n},\frac{\gamma\delta}{\alpha}q,\frac{\gamma}{% \beta}q,\delta q;q\right)_{\infty}}.$

Duality

 18.28.23 $R_{n}\left(q^{-y}+\gamma\delta q^{y+1};\alpha,\beta,\gamma,\delta\,|\,q\right)% =R_{y}\left(q^{-n}+\alpha\beta q^{n+1};\gamma,\delta,\alpha,\beta\,|\,q\right),$ $\alpha q$, $\beta\delta q$, or $\gamma q=q^{-N}$; $n,y=0,1,\ldots,N$.

Leonard (1982) classified all (finite or infinite) discrete systems of OP’s $p_{n}(x)$ on a set $\{x(m)\}$ for which there is a system of discrete OP’s $q_{m}(y)$ on a set $\{y(n)\}$ such that $p_{n}(x(m))=q_{m}(y(n))$. These systems are the $q$-Racah polynomials and its limit cases.

§18.28(ix) Continuous $q$-Jacobi Polynomials

Let $x=\tfrac{1}{2}(z+z^{-1})$ and let $R_{n}(z)$ be given by (18.28.1_5). The continuous $q$-Jacobi polynomial $P_{n}^{(\alpha,\beta)}(x\,|\,q)$ is defined by

 18.28.24 $\frac{\left(q;q\right)_{n}}{\left(q^{\alpha+1};q\right)_{n}}P_{n}^{(\alpha,% \beta)}(x\,|\,q)=R_{n}\left(z;q^{\frac{1}{2}\alpha+\frac{1}{4}},q^{\frac{1}{2}% \alpha+\frac{3}{4}},-q^{\frac{1}{2}\beta+\frac{1}{4}},-q^{\frac{1}{2}\beta+% \frac{3}{4}}\,|\,q\right)={{}_{4}\phi_{3}}\left({q^{-n},q^{n+\alpha+\beta+1},q% ^{\frac{1}{2}\alpha+\frac{1}{4}}z,q^{\frac{1}{2}\alpha+\frac{1}{4}}z^{-1}\atop q% ^{\alpha+1},-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}% ;q,q\right).$

Specialization to continuous $q$-ultraspherical:

 18.28.25 $P_{n}^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x\,|\,q)=\frac{q^{\frac{1}{2% }n\lambda}\left(q^{\lambda+\ifrac{1}{2}};q\right)_{n}}{\left(q^{2\lambda};q% \right)_{n}}C_{n}\left(x;q^{\lambda}\,|\,q\right).$

§18.28(x) Limit Relations

Let $R_{n}(z)$ be as defined in (18.28.1_5) and put $r_{n}(x;a,b,c,d\,|\,q)=R_{n}(z;a,b,c,d\,|\,q)$, $x=\frac{1}{2}(z+z^{-1})$.

From Askey–Wilson to Big $q$-Jacobi

 18.28.26 $\lim_{\lambda\to 0}r_{n}\left(\ifrac{x}{(2\lambda)};\lambda,qa\lambda^{-1},qc% \lambda^{-1},bc^{-1}\lambda\,|\,q\right)=P_{n}\left(x;a,b,c;q\right).$ ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c};\NVar{q}\right)$: big $q$-Jacobi polynomial, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: multiple $q$-Pochhammer symbol, $z$: complex variable, $q$: real variable, $\ell$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Source: Koekoek et al. (2010, (14.1.18)) Proof sketch: In the terminating $q$-hypergeometric series which defines the ${}_{4}\phi_{3}$ in (18.28.1_5), with $\left(azq^{\ell},az^{-1}q^{\ell};q\right)_{\ell}$ written as $\prod_{j=0}^{\ell-1}(1-2aq^{j}x+a^{2}q^{2j})$, substitute for $x$ and the parameters according to the left-hand side of the present formula, take the limit, and obtain the ${}_{3}\phi_{2}$ as in (18.27.5). Referenced by: (18.28.27), (18.28.28), §18.28(x) Permalink: http://dlmf.nist.gov/18.28.E26 Encodings: TeX, pMML, png See also: Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

From Askey–Wilson to Little $q$-Jacobi

 18.28.27 $\lim_{\lambda\to 0}r_{n}\left(\ifrac{bqx}{(2\lambda)};\lambda,qb\lambda^{-1},q% ,a\,|\,q\right)=(-b)^{n}q^{\ifrac{n(n+1)}{2}}\frac{\left(qa;q\right)_{n}}{% \left(qb;q\right)_{n}}p_{n}\left(x;a,b;q\right).$ ⓘ Symbols: $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b};\NVar{q}\right)$: little $q$-Jacobi polynomial, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $n$: nonnegative integer and $x$: real variable Proved: Koornwinder (1993, (6.4)); The limit formula there is equivalent to the present formula(proved) Proof sketch: Proceed in a similar way as in the proof of (18.28.26). After taking the limit one now obtains a ${}_{3}\phi_{2}$ which equals (18.27.14_1) by (18.27.13). Referenced by: §18.28(x) Permalink: http://dlmf.nist.gov/18.28.E27 Encodings: TeX, pMML, png See also: Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18
 18.28.28 $\lim_{\mu\to 0,\;\ifrac{\lambda}{\mu}\to 0}r_{n}\left(\ifrac{x}{(2\lambda\mu)}% ;\ifrac{\lambda}{\mu},\ifrac{qa\mu}{\lambda},\ifrac{1}{(\lambda\mu)},qb\lambda% \mu\,|\,q\right)=p_{n}\left(x;a,b;q\right).$ ⓘ Symbols: $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b};\NVar{q}\right)$: little $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable Proof sketch: Proceed in a similar way as in the proof of (18.28.26). After taking the limit one now obtains a ${}_{2}\phi_{1}$ which equals (18.27.13). Referenced by: §18.28(x) Permalink: http://dlmf.nist.gov/18.28.E28 Encodings: TeX, pMML, png See also: Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

 18.28.29 $\lim_{q\to 1}\frac{p_{n}\left(1-\tfrac{1}{2}x(1-q)^{2};q^{a},q^{b},q^{c},q^{d}% \,|\,q\right)}{(1-q)^{3n}}=W_{n}\left(x;a,b,c,d\right).$ ⓘ Symbols: $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\,|\,\NVar{q}\right)$: Askey–Wilson polynomial, $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$: Wilson polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable Proof sketch: Substitute (18.28.1) in the left-hand side, take the limit, and use (18.26.1). Notes: Another form of this limit is given in Koekoek et al. (2010, (14.1.21)). Referenced by: §18.28(x) Permalink: http://dlmf.nist.gov/18.28.E29 Encodings: TeX, pMML, png See also: Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

From Continuous $q$-Jacobi to Jacobi

 18.28.30 $\lim_{q\to 1}P_{n}^{(\alpha,\beta)}(x\,|\,q)=P^{(\alpha,\beta)}_{n}\left(x% \right).$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $(\NVar{a},\NVar{b})$: open interval, $q$: real variable, $n$: nonnegative integer and $x$: real variable Source: Koekoek et al. (2010, (14.10.15)) Referenced by: §18.28(x) Permalink: http://dlmf.nist.gov/18.28.E30 Encodings: TeX, pMML, png See also: Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

From Continuous $q$-Ultraspherical to Ultraspherical

 18.28.31 $\lim_{q\to 1}C_{n}\left(x;q^{\lambda}\,|\,q\right)=C^{(\lambda)}_{n}\left(x% \right).$ ⓘ Symbols: $C_{\NVar{n}}\left(\NVar{x};\NVar{\beta}\,|\,\NVar{q}\right)$: continuous $q$-ultraspherical polynomial, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$: ultraspherical (or Gegenbauer) polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable Source: Koekoek et al. (2010, (14.10.35)) Referenced by: §18.28(x) Permalink: http://dlmf.nist.gov/18.28.E31 Encodings: TeX, pMML, png See also: Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

From Continuous $q$-Ultraspherical to Continuous $q$-Hermite

 18.28.32 $\lim_{\beta\to 0}C_{n}\left(x;\beta\,|\,q\right)=\frac{H_{n}\left(x\,|\,q% \right)}{\left(q;q\right)_{n}}.$

From Continuous $q$-Hermite to Hermite

 18.28.33 $\lim_{q\to 1}\frac{H_{n}\left(x\sqrt{\ifrac{(1-q)}{2}}\,|\,q\right)}{(\ifrac{(% 1-q)}{2})^{\ifrac{n}{2}}}=H_{n}\left(x\right).$ ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $H_{\NVar{n}}\left(\NVar{x}\,|\,\NVar{q}\right)$: continuous $q$-Hermite polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable Source: Koekoek et al. (2010, (14.26.14)) Referenced by: §18.28(x) Permalink: http://dlmf.nist.gov/18.28.E33 Encodings: TeX, pMML, png See also: Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

From $q$-Racah to Racah

 18.28.34 $\lim_{q\to 1}R_{n}\left(q^{-y}+q^{y+\gamma+\delta+1};q^{\alpha},q^{\beta},q^{% \gamma},q^{\delta}\,|\,q\right)=R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,% \gamma,\delta\right).$

§18.28(xi) Limits for $q\downarrow-1$

Bannai and Ito (1984) introduced OP’s, called the Bannai–Ito polynomials which are the limit for $q\downarrow-1$ of the $q$-Racah polynomials. They have to be included in the classification by Leonard (1982), mentioned in §18.28(viii). In Tsujimoto et al. (2012) an extension of the Bannai–Ito polynomials occurs as eigenfunctions of a Dunkl type operator. Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).