§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 (2005, Chapter 15), and Koornwinder (2012). For the notation of $q$-hypergeometric functions see §§17.2 and 17.4(i).

 18.28.1 $p_{n}(\mathop{\cos\/}\nolimits\theta)=\mathop{p_{n}\/}\nolimits\!\left(\mathop% {\cos\/}\nolimits\theta;a,b,c,d\,|\,q\right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}% \left(abq^{\ell},acq^{\ell},adq^{\ell};q\right)_{n-\ell}\*\frac{\left(q^{-n},% abcdq^{n-1};q\right)_{\ell}}{\left(q;q\right)_{\ell}}\prod_{j=0}^{\ell-1}{(1-2% aq^{j}\mathop{\cos\/}\nolimits\theta+a^{2}q^{2j})}=a^{-n}\left(ab,ac,ad;q% \right)_{n}\*\mathop{{{}_{4}\phi_{3}}\/}\nolimits\!\left({q^{-n},abcdq^{n-1},% ae^{i\theta},ae^{-i\theta}\atop ab,ac,ad};q,q\right).$

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. Furthermore, $|ab|$, $|ac|$, $|ad|$, $|bc|$, $|bd|$, $|cd|<1$. Then

 18.28.2 $\int_{-1}^{1}p_{n}(x)p_{m}(x)w(x)dx=h_{n}\delta_{n,m},$ $|a|,|b|,|c|,|d|\leq 1$,

where

 18.28.3 $2\pi\mathop{\sin\/}\nolimits\theta\,w(\mathop{\cos\/}\nolimits\theta)=\left|% \frac{\left(e^{2i\theta};q\right)_{\infty}}{\left(ae^{i\theta},be^{i\theta},ce% ^{i\theta},de^{i\theta};q\right)_{\infty}}\right|^{2},$
 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 $h_{n}=h_{0}\frac{(1-abcdq^{n-1})\left(q,ab,ac,ad,bc,bd,cd;q\right)_{n}}{(1-% abcdq^{2n-1})\left(abcd;q\right)_{n}},$ $n=1,2,\ldots$.

More generally, without the constraints in (18.28.2),

 18.28.6 $\int_{-1}^{1}p_{n}(x)p_{m}(x)w(x)dx+\sum_{\ell}p_{n}(x_{\ell})p_{m}(x_{\ell})% \omega_{\ell}=h_{n}\delta_{n,m},$

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 $\alpha=a$.

§18.28(iii) Al-Salam–Chihara Polynomials

 18.28.7 $\mathop{Q_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta;a,b\,|\,q% \right)=\mathop{p_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta;a,b,0,% 0\,|\,q\right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}\frac{\left(abq^{\ell};q\right)_% {n-\ell}\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}\*\prod_{j=0}^{% \ell-1}(1-2aq^{j}\mathop{\cos\/}\nolimits\theta+a^{2}q^{2j})=\frac{\left(ab;q% \right)_{n}}{a^{n}}\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({q^{-n},ae^{i% \theta},ae^{-i\theta}\atop ab,0};q,q\right)=\left(be^{-i\theta};q\right)_{n}e^% {in\theta}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({q^{-n},ae^{i\theta}% \atop b^{-1}q^{1-n}e^{i\theta}};q,b^{-1}qe^{-i\theta}\right).$
 18.28.8 $\frac{1}{2\pi}\int_{0}^{\pi}\mathop{Q_{n}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta;a,b\,|\,q\right)\mathop{Q_{m}\/}\nolimits\!\left(\mathop{\cos% \/}\nolimits\theta;a,b\,|\,q\right)\*\left|\frac{\left(e^{2i\theta};q\right)_{% \infty}}{\left(ae^{i\theta},be^{i\theta};q\right)_{\infty}}\right|^{2}d\theta=% \frac{\delta_{n,m}}{\left(q^{n+1},abq^{n};q\right)_{\infty}},$ $a,b\in\Real$ or $a=\conj{b}$; $|ab|<1$; $|a|,|b|\leq 1$.

More generally, without the constraints $|a|,|b|\leq 1$ 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 $\mathop{Q_{n}\/}\nolimits\!\left(\tfrac{1}{2}(aq^{-y}+a^{-1}q^{y});a,b\,|\,q^{% -1}\right)=(-1)^{n}b^{n}q^{-\frac{1}{2}n(n-1)}\*\left((ab)^{-1};q\right)_{n}% \mathop{{{}_{3}\phi_{1}}\/}\nolimits\!\left({q^{-n},q^{-y},a^{-2}q^{y}\atop(ab% )^{-1}};q,q^{n}ab^{-1}\right).$ Defines: $\mathop{Q_{n}\/}\nolimits\!\left(x;a,b\,|\,q^{-1}\right)$: $q^{-1}$-Al-Salam–Chihara polynomial Symbols: $\left(a;q\right)_{n}$: $q$-factorial (or $q$-shifted factorial), $\mathop{{{}_{r+1}\phi_{s}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_% {1},b_{2},\dots,b_{s}};q,z\right)$: 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
 18.28.10 $\sum_{y=0}^{\infty}\frac{(1-q^{2y}a^{-2})\left(a^{-2},(ab)^{-1};q\right)_{y}}{% (1-a^{-2})\left(q,bqa^{-1};q\right)_{y}}(ba^{-1})^{y}q^{y^{2}}\*\mathop{Q_{n}% \/}\nolimits\!\left(\tfrac{1}{2}(aq^{-y}+a^{-1}q^{y});a,b\,|\,q^{-1}\right)\*% \mathop{Q_{m}\/}\nolimits\!\left(\tfrac{1}{2}(aq^{-y}+a^{-1}q^{y});a,b\,|\,q^{% -1}\right)=\frac{\left(qa^{-2};q\right)_{\infty}}{\left(ba^{-1}q;q\right)_{% \infty}}\left(q,(ab)^{-1};q\right)_{n}(ab)^{n}q^{-n^{2}}\delta_{n,m}.$

Eq. (18.28.10) is valid when either

 18.28.11 $01,a^{-1}b Symbols: $\in$: element of, $\Real$: real line (excluding infinity) and $q$: real variable Referenced by: §18.28(iv) Permalink: http://dlmf.nist.gov/18.28.E11 Encodings: TeX, pMML, png

or

 18.28.12 $00,a^{-1}b%

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 uniquely determined. 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 $\displaystyle\mathop{C_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta;% \beta\,|\,q\right)$ $\displaystyle=\sum_{\ell=0}^{n}\frac{\left(\beta;q\right)_{\ell}\left(\beta;q% \right)_{n-\ell}}{\left(q;q\right)_{\ell}\left(q;q\right)_{n-\ell}}e^{i(n-2% \ell)\theta}$ $\displaystyle=\frac{\left(\beta;q\right)_{n}}{\left(q;q\right)_{n}}e^{in\theta% }\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({q^{-n},\beta\atop\beta^{-1}q^{1-% n}};q,\beta^{-1}qe^{-2i\theta}\right).$ Defines: $\mathop{C_{n}\/}\nolimits\!\left(x;\beta\,|\,q\right)$: continuous $q$-ultraspherical polynomial Symbols: $\mathop{\cos\/}\nolimits z$: cosine function, $e$: base of exponential function, $\left(a;q\right)_{n}$: $q$-factorial (or $q$-shifted factorial), $\mathop{{{}_{r+1}\phi_{s}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_% {1},b_{2},\dots,b_{s}};q,z\right)$: 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 18.28.14 $\displaystyle\mathop{C_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta;% \beta\,|\,q\right)$ $\displaystyle=\frac{\left(\beta^{2};q\right)_{n}}{\left(q;q\right)_{n}\beta^{% \frac{1}{2}n}}\mathop{{{}_{4}\phi_{3}}\/}\nolimits\!\left({q^{-n},\beta^{2}q^{% n},\beta^{\frac{1}{2}}e^{i\theta},\beta^{\frac{1}{2}}e^{-i\theta}\atop\beta q^% {\frac{1}{2}},-\beta,-\beta q^{\frac{1}{2}}};q,q\right).$
 18.28.15 $\frac{1}{2\pi}\int_{0}^{\pi}\mathop{C_{n}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta;\beta\,|\,q\right)\mathop{C_{m}\/}\nolimits\!\left(\mathop{% \cos\/}\nolimits\theta;\beta\,|\,q\right)\*\left|\frac{\left(e^{2i\theta};q% \right)_{\infty}}{\left(\beta e^{2i\theta};q\right)_{\infty}}\right|^{2}d% \theta=\frac{\left(\beta,\beta q;q\right)_{\infty}}{\left(\beta^{2},q;q\right)% _{\infty}}\frac{(1-\beta)\left(\beta^{2};q\right)_{n}}{(1-\beta q^{n})\left(q;% q\right)_{n}}\delta_{n,m},$ $-1<\beta<1$.

These polynomials are also called Rogers polynomials.

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

 18.28.16 $\mathop{H_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\,|\,q\right)=% \sum_{\ell=0}^{n}\frac{\left(q;q\right)_{n}e^{i(n-2\ell)\theta}}{\left(q;q% \right)_{\ell}\left(q;q\right)_{n-\ell}}=e^{in\theta}\mathop{{{}_{2}\phi_{0}}% \/}\nolimits\!\left({q^{-n},0\atop-};q,q^{n}e^{-2i\theta}\right).$ Defines: $\mathop{H_{n}\/}\nolimits\!\left(x\,|\,q\right)$: continuous $q$-Hermite polynomial Symbols: $\mathop{\cos\/}\nolimits z$: cosine function, $e$: base of exponential function, $\left(a;q\right)_{n}$: $q$-factorial (or $q$-shifted factorial), $\mathop{{{}_{r+1}\phi_{s}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_% {1},b_{2},\dots,b_{s}};q,z\right)$: 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.E16 Encodings: TeX, pMML, png
 18.28.17 $\frac{1}{2\pi}\int_{0}^{\pi}\mathop{H_{n}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\,|\,q\right)\mathop{H_{m}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\,|\,q\right)\left|\left(e^{2i\theta};q\right)_{\infty}\right|^% {2}d\theta=\frac{\delta_{n,m}}{\left(q^{n+1};q\right)_{\infty}}.$

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

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

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 $R_{n}(x)=\mathop{R_{n}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\,|\,q% \right)=\sum_{\ell=0}^{n}\frac{q^{\ell}\left(q^{-n},\alpha\beta q^{n+1};q% \right)_{\ell}}{\left(\alpha q,\beta\delta q,\gamma q,q;q\right)_{\ell}}\*% \prod_{j=0}^{\ell-1}(1-q^{j}x+\gamma\delta q^{2j+1})=\mathop{{{}_{4}\phi_{3}}% \/}\nolimits\!\left({q^{-n},\alpha\beta q^{n+1},q^{-y},\gamma\delta q^{y+1}% \atop\alpha q,\beta\delta q,\gamma q};q,q\right),$ $\alpha q$, $\beta\delta q$, or $\gamma q=q^{-N}$; $n=0,1,\dots,N$. Defines: $\mathop{R_{n}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)$: $q$-Racah polynomial Symbols: $\left(a;q\right)_{n}$: $q$-factorial (or $q$-shifted factorial), $\mathop{{{}_{r+1}\phi_{s}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_% {1},b_{2},\dots,b_{s}};q,z\right)$: basic hypergeometric (or $q$-hypergeometric) function, $y$: real variable, $q$: real variable, $\ell$: nonnegative integer, $n$: nonnegative integer, $N$: positive integer, $\delta$: arbitary small positive constant, $x$: real variable and $R_{n}(x)$ Permalink: http://dlmf.nist.gov/18.28.E19 Encodings: TeX, pMML, png
 18.28.20 $\sum_{y=0}^{N}R_{n}(q^{-y}+\gamma\delta q^{y+1})R_{m}(q^{-y}+\gamma\delta q^{y% +1})\omega_{y}=h_{n}\delta_{n,m},$ $n,m=0,1,\dots,N$.

For $\omega_{y}$ and $h_{n}$ see Koekoek et al. (2010, Eq. (14.2.2)).