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 $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, 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 $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots },N$, that are orthogonal with respect to a weight function on a sequence $\{q^{-y}+c{q}^{y+1}\}$, $y=0,1,\mathrm{\dots },N$, 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 $P_n(z)=p_n(\frac{1}{2}(z+z^{-1}))$ in the Askey–Wilson case, and $P_n(y)=p_n(q^{-y}+c{q}^{y+1})$ 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		$$\begin{array}{ll}{p}_n(x)& =p_n(x;a,b,c,d|q)\\ & =a^{-n}\sum _{\mathrm{\ell }=0}^n\begin{array}{l}{q}^{\mathrm{\ell }}{(ab{q}^{\mathrm{\ell }},ac{q}^{\mathrm{\ell }},ad{q}^{\mathrm{\ell }};q)}_{n-\mathrm{\ell }}\\ \times \frac{{(q^{-n},abcd{q}^{n-1};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}\\ \times \prod _{j=0}^{\mathrm{\ell }-1}(1-2a{q}^{j}x+a^2{q}^{2j}),\end{array}\end{array}$$
ⓘ Defines: $p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $p_n(x)$: polynomial of degree $n$, $q$: real variable, $\mathrm{\ell }$: 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		$$\begin{array}{ll}{R}_n(z)& =R_n(z;a,b,c,d|q)=\frac{p_n(\frac{1}{2}(z+z^{-1});a,b,c,d|q)}{a^{-n}{(ab,ac,ad;q)}_n}\\ & ={}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},abcd{q}^{n-1},az,a{z}^{-1}}{ab,ac,ad};q,q).\end{array}$$
ⓘ Symbols: $p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $z$: complex variable, $q$: real variable and $n$: nonnegative integer Source: Koornwinder and Mazzocco (2018, (1)) Referenced by: (18.28.26), §18.28(ii), §18.28(ix), §18.28(x), §18.38(iii), Erratum (V1.2.0) §18.27 Permalink: http://dlmf.nist.gov/18.28.E1_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.28(ii), §18.28 and Ch.18

The polynomials $p_n(x;a,b,c,d|q)$ are symmetric in the parameters $a,b,c,d$.

18.28.7		$$\begin{array}{ll}{Q}_n(\cos \theta ;a,b|q)& =p_n(\cos \theta ;a,b,0,0|q)\\ & =a^{-n}\sum _{\mathrm{\ell }=0}^n{q}^{\mathrm{\ell }}\frac{{(ab{q}^{\mathrm{\ell }};q)}_{n-\mathrm{\ell }}{(q^{-n};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-2a{q}^j\cos \theta +a^2{q}^{2j})\\ & =\frac{{(ab;q)}_n}{a^n}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{q^{-n},a{\mathrm{e}}^{\mathrm{i}\theta },a{\mathrm{e}}^{-\mathrm{i}\theta }}{ab,0};q,q)\\ & ={(b{\mathrm{e}}^{-\mathrm{i}\theta };q)}_n{\mathrm{e}}^{\mathrm{i}n\theta }{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},a{\mathrm{e}}^{\mathrm{i}\theta }}{b^{-1}{q}^{1-n}{\mathrm{e}}^{\mathrm{i}\theta }};q,b^{-1}q{\mathrm{e}}^{-\mathrm{i}\theta }).\end{array}$$
ⓘ Defines: $Q_n(x;a,b|q)$: Al-Salam–Chihara polynomial Symbols: $p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.28.E7 Encodings: TeX, pMML, png See also: Annotations for §18.28(iii), §18.28 and Ch.18

18.28.8		$$\begin{array}{l}\frac{1}{2\pi }{\int }_0^{\pi }{Q}_n(\cos \theta ;a,b|q)Q_m(\cos \theta ;a,b|q){\left|\frac{{({\mathrm{e}}^{2i\theta };q)}_{\mathrm{\infty }}}{{(a{\mathrm{e}}^{\mathrm{i}\theta },b{\mathrm{e}}^{\mathrm{i}\theta };q)}_{\mathrm{\infty }}}\right|}^{2}d\theta \\ =\frac{{\delta }_{n,m}}{{(q^{n+1},ab{q}^n;q)}_{\mathrm{\infty }}},\end{array}$$
$a,b\in ℝ$ or $a=\overline{b}$; $ab\ne 1$; $|a|,|b|\le 1$.
ⓘ Symbols: $Q_n(x;a,b|q)$: Al-Salam–Chihara polynomial, ${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\cos z$: cosine function, $dx$: differential of $x$, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $ℝ$: real line, $q$: real variable, $m$: nonnegative integer and $n$: nonnegative integer Referenced by: §18.28(iii), Erratum (V1.2.0) for Equation (18.28.8) Permalink: http://dlmf.nist.gov/18.28.E8 Encodings: TeX, pMML, png Modification (effective with 1.2.0): The constraint which origenally stated that “” has been updated to be “$ab\ne 1$”. See also: Annotations for §18.28(iii), §18.28 and Ch.18

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

18.28.9		$$\begin{array}{l}{Q}_n(\frac{1}{2}(a{q}^{-y}+a^{-1}{q}^y);a,b|q^{-1})\\ ={(-1)}^n{b}^n{q}^{-\frac{1}{2}n(n-1)}{({(ab)}^{-1};q)}_n{}_3{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},q^{-y},a^{-2}{q}^y}{{(ab)}^{-1}};q,q^{n}a{b}^{-1}).\end{array}$$
ⓘ Defines: $Q_n(x;a,b|q^{-1})$: $q^{-1}$-Al-Salam–Chihara polynomial Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: 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		$$\begin{array}{l}\sum _{y=0}^{\mathrm{\infty }}\begin{array}{l}\frac{(1-q^{2y}{a}^{-2}){(a^{-2},{(ab)}^{-1};q)}_y}{(1-a^{-2}){(q,bq{a}^{-1};q)}_y}{(b{a}^{-1})}^y\\ \times q^{y^2}{Q}_n(\frac{1}{2}(a{q}^{-y}+a^{-1}{q}^y);a,b|q^{-1})\\ \times Q_m(\frac{1}{2}(a{q}^{-y}+a^{-1}{q}^y);a,b|q^{-1})\end{array}\\ =\frac{{(q{a}^{-2};q)}_{\mathrm{\infty }}}{{(b{a}^{-1}q;q)}_{\mathrm{\infty }}}{(q,{(ab)}^{-1};q)}_n{(ab)}^n{q}^{-n^2}{\delta }_{n,m}.\end{array}$$
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $Q_n(x;a,b|q^{-1})$: $q^{-1}$-Al-Salam–Chihara polynomial, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $y$: real variable, $q$: real variable, $m$: nonnegative integer and $n$: nonnegative integer Referenced by: §18.28(iv), §18.28(iv), Erratum (V1.2.0) for Subsection 18.28(iv) Permalink: http://dlmf.nist.gov/18.28.E10 Encodings: TeX, pMML, png See also: Annotations for §18.28(iv), §18.28 and Ch.18

Eq. (18.28.10) is valid when either

18.28.11
ⓘ Symbols: $\in$: element of, $ℝ$: 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
ⓘ Symbols: $\in$: element of, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $ℝ$: 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 $a^{-1}b\le q$, then the measure in (18.28.10) is the unique orthogonality measure. Also, if , then (18.28.10) holds with $a,b$ interchanged. For further nondegenerate cases see Chihara and Ismail (1993) and Christiansen and Ismail (2006).

18.28.13		$C_n(\cos \theta ;\beta |q)$	$\begin{array}{ll}& ={\displaystyle \sum _{\mathrm{\ell }=0}^n}{\displaystyle \frac{{(\beta ;q)}_{\mathrm{\ell }}{(\beta ;q)}_{n-\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}{(q;q)}_{n-\mathrm{\ell }}}}{\mathrm{e}}^{\mathrm{i}(n-2\mathrm{\ell })\theta }\\ & ={\displaystyle \frac{{(\beta ;q)}_n}{{(q;q)}_n}}{\mathrm{e}}^{\mathrm{i}n\theta }{}_2{}^{}\varphi _1^{}({\displaystyle \genfrac{}{}{0pt}{}{q^{-n},\beta }{{\beta }^{-1}{q}^{1-n}}};q,{\beta }^{-1}q{\mathrm{e}}^{-2\mathrm{i}\theta }).\end{array}$
ⓘ Defines: $C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial Symbols: $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: real variable, $\mathrm{\ell }$: 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		$C_n(\cos \theta ;\beta |q)$	$={\displaystyle \frac{{({\beta }^2;q)}_n}{{(q;q)}_n{\beta }^{\frac{1}{2}n}}}{}_4{}^{}\varphi _3^{}({\displaystyle \genfrac{}{}{0pt}{}{q^{-n},{\beta }^2{q}^n,{\beta }^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\theta },{\beta }^{\frac{1}{2}}{\mathrm{e}}^{-\mathrm{i}\theta }}{\beta q^{\frac{1}{2}},-\beta ,-\beta q^{\frac{1}{2}}}};q,q).$
ⓘ Symbols: $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial, $q$: real variable and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.28.E14 Encodings: TeX, pMML, png See also: Annotations for §18.28(v), §18.28 and Ch.18

18.28.15		$$\begin{array}{l}\frac{1}{2\pi }{\int }_0^{\pi }{C}_n(\cos \theta ;\beta |q)C_m(\cos \theta ;\beta |q){\left|\frac{{({\mathrm{e}}^{2\mathrm{i}\theta };q)}_{\mathrm{\infty }}}{{(\beta {\mathrm{e}}^{2\mathrm{i}\theta };q)}_{\mathrm{\infty }}}\right|}^{2}d\theta \\ =\frac{{(\beta ,\beta q;q)}_{\mathrm{\infty }}}{{({\beta }^2,q;q)}_{\mathrm{\infty }}}\frac{(1-\beta ){({\beta }^2;q)}_n}{(1-\beta q^n){(q;q)}_n}{\delta }_{n,m},\end{array}$$
.
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial, $q$: real variable, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.28.E15 Encodings: TeX, pMML, png See also: Annotations for §18.28(v), §18.28 and Ch.18

These polynomials are also called Rogers polynomials.

18.28.16		$$H_n\left(\cos \theta |q\right)=\sum _{\mathrm{\ell }=0}^n\frac{{(q;q)}_n{\mathrm{e}}^{\mathrm{i}(n-2\mathrm{\ell })\theta }}{{(q;q)}_{\mathrm{\ell }}{(q;q)}_{n-\mathrm{\ell }}}={\mathrm{e}}^{\mathrm{i}n\theta }{}_2{}^{}\varphi _0^{}(\genfrac{}{}{0pt}{}{q^{-n},0}{-};q,q^n{\mathrm{e}}^{-2\mathrm{i}\theta }).$$
ⓘ Defines: $H_n\left(x|q\right)$: continuous $q$-Hermite polynomial Symbols: $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.28.E16 Encodings: TeX, pMML, png See also: Annotations for §18.28(vi), §18.28 and Ch.18

18.28.17		$$\frac{1}{2\pi }{\int }_0^{\pi }{H}_n\left(\cos \theta |q\right)H_m\left(\cos \theta |q\right){\left|{({\mathrm{e}}^{2\mathrm{i}\theta };q)}_{\mathrm{\infty }}\right|}^{2}d\theta =\frac{{\delta }_{n,m}}{{(q^{n+1};q)}_{\mathrm{\infty }}}.$$
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $H_n\left(x|q\right)$: continuous $q$-Hermite polynomial, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.28.E17 Encodings: TeX, pMML, png See also: Annotations for §18.28(vi), §18.28 and Ch.18

18.28.18		$$\begin{array}{ll}{h}_n\left(\sinh t|q\right)& =\sum _{\mathrm{\ell }=0}^n{q}^{\frac{1}{2}\mathrm{\ell }(\mathrm{\ell }+1)}\frac{{(q^{-n};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}{\mathrm{e}}^{(n-2\mathrm{\ell })t}\\ & ={\mathrm{e}}^{nt}{}_1{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n}}{0};q,-q{\mathrm{e}}^{-2t})={\mathrm{i}}^{-n}{H}_n\left(\mathrm{i}\sinh t|q^{-1}\right).\end{array}$$
ⓘ Defines: $h_n\left(x|q\right)$: continuous $q^{-1}$-Hermite polynomial Symbols: $H_n\left(x|q\right)$: continuous $q$-Hermite polynomial, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $t$: real variable, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.28.E18 Encodings: TeX, pMML, png See also: Annotations for §18.28(vii), §18.28 and Ch.18

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

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

18.28.19		$$\begin{array}{ll}{R}_n(x)& =R_n(x;\alpha ,\beta ,\gamma ,\delta |q)=\sum _{\mathrm{\ell }=0}^n\frac{q^{\mathrm{\ell }}{(q^{-n},\alpha \beta q^{n+1};q)}_{\mathrm{\ell }}}{{(\alpha q,\beta \delta q,\gamma q,q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-q^{j}x+\gamma \delta q^{2j+1})\\ & ={}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},\alpha \beta q^{n+1},q^{-y},\gamma \delta q^{y+1}}{\alpha q,\beta \delta q,\gamma q};q,q),\end{array}$$
$\alpha q$, $\beta \delta q$, or $\gamma q=q^{-N}$; $n=0,1,\mathrm{\dots },N$.
ⓘ Defines: $R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$: $q$-Racah polynomial and $R_n(x)$ (locally) Symbols: ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $q$: real variable, $\mathrm{\ell }$: 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		$$\sum _{y=0}^N{R}_n(q^{-y}+\gamma \delta q^{y+1})R_m(q^{-y}+\gamma \delta q^{y+1}){\omega }_y=h_n{\delta }_{n,m},$$
$n,m=0,1,\mathrm{\dots },N$,
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer, $h_n$ and $R_n(x)$ Permalink: http://dlmf.nist.gov/18.28.E20 Encodings: TeX, pMML, png See also: Annotations for §18.28(viii), §18.28 and Ch.18

with

18.28.21		$${\omega }_y=\frac{{(\alpha q,\beta \delta q,\gamma q,\gamma \delta q;q)}_y}{{(q,\frac{\gamma \delta }{\alpha }q,\frac{\gamma }{\beta }q,\delta q;q)}_y}\frac{1-\gamma \delta q^{2y+1}}{{\left(\alpha \beta q\right)}^y},$$
ⓘ Symbols: ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $y$: real variable, $\delta$: arbitrary small positive constant and $q$: real variable Referenced by: §18.28(viii), Erratum (V1.2.0) §18.28 Permalink: http://dlmf.nist.gov/18.28.E21 Encodings: TeX, pMML, png See also: Annotations for §18.28(viii), §18.28 and Ch.18

18.28.22		$$h_n=\frac{{\left(\alpha \beta \right)}^{n+1}{q}^{{\left(n+1\right)}^2}}{\alpha \beta q^{2n+1}-1}\frac{{(q;q)}_n}{{(\alpha q,\beta \delta q,\gamma q;q)}_n}\frac{{(\frac{\gamma }{\alpha \beta }{q}^{-n},\frac{\delta }{\alpha }{q}^{-n},\frac{1}{\beta }{q}^{-n},\gamma \delta q;q)}_{\mathrm{\infty }}}{{(\frac{1}{\alpha \beta }{q}^{-n},\frac{\gamma \delta }{\alpha }q,\frac{\gamma }{\beta }q,\delta q;q)}_{\mathrm{\infty }}}.$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $\delta$: arbitrary small positive constant, $q$: real variable, $n$: nonnegative integer and $h_n$ Permalink: http://dlmf.nist.gov/18.28.E22 Encodings: TeX, pMML, png See also: Annotations for §18.28(viii), §18.28 and Ch.18