18 Orthogonal Polynomials Other Orthogonal Polynomials 18.27

q

-Hahn Class 18.29 Asymptotic Approximations for

q

-Hahn and Askey–Wilson Classes

§18.28 Askey–Wilson Class

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Keywords:: Askey–Wilson class orthogonal polynomials
Referenced by:: §18.27(i), Erratum (V1.2.0) §18.28
Permalink:: http://dlmf.nist.gov/18.28
See also:: Annotations for Ch.18

§18.28(i) Introduction
§18.28(ii) Askey–Wilson Polynomials
§18.28(iii) Al-Salam–Chihara Polynomials
§18.28(iv) $q^{-1}$ -Al-Salam–Chihara Polynomials
§18.28(v) Continuous $q$ -Ultraspherical Polynomials
§18.28(vi) Continuous $q$ -Hermite Polynomials
§18.28(vii) Continuous $q^{-1}$ -Hermite Polynomials
§18.28(viii) $q$ -Racah Polynomials
§18.28(ix) Continuous $q$ -Jacobi Polynomials
§18.28(x) Limit Relations
§18.28(xi) Limits for $q\downarrow-1$

§18.28(i) Introduction

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Keywords:: Askey–Wilson class orthogonal polynomials, as eigenfunctions of a $q$ -difference operator, orthogonality properties, representation as $q$ -hypergeometric functions
Referenced by:: Erratum (V1.0.7) for References
Permalink:: http://dlmf.nist.gov/18.28.i
Addition (effective with 1.0.7):: The citation to Koornwinder (2012) has been added at the end of this subsection.
See also:: Annotations for §18.28 and Ch.18

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(ii) Askey–Wilson Polynomials

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Keywords:: Askey–Wilson class orthogonal polynomials, Askey–Wilson polynomials, $q$ -hypergeometric function, relation to $q$ -hypergeometric functions, relations to other functions
Notes:: See Askey and Wilson (1985), Gasper and Rahman (2004, (7.5.2), (7.5.15), (7.5.21)), Ismail (2009, (15.2.4), (15.2.5)).
Referenced by:: Erratum (V1.2.0) §18.28
Permalink:: http://dlmf.nist.gov/18.28.ii
Addition (effective with 1.2.0):: Equation (18.28.1) was updated such that $\cos\theta$ was replaced with $x$ . Equations (18.28.1_5), (18.28.6_1)–(18.28.6_5) were added. The ${{}_{4}\phi_{3}}$ representation which was originally in (18.28.1) was moved to the new equation (18.28.1_5) in a normalized fashion. A sentence was added underneath (18.28.1_5) explaining the symmetry of the Askey–Wilson polynomials in terms of its parameters $a, b, c, d$ .
See also:: Annotations for §18.28 and Ch.18

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		$R_{n}(z)=R_{n}(z;a,b,c,d\,\|\,q)=\frac{p_{n}\left(\frac{1}{2}(z+z^{-1});a,b,c,d% \,\|\,q\right)}{a^{-n}\left(ab,ac,ad;q\right)_{n}}={{}_{4}\phi_{3}}\left({q^{-n% },abcdq^{n-1},az,az^{-1}\atop ab,ac,ad};q,q\right).$
ⓘ Symbols: $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\,\|\,\NVar{q}\right)$ : Askey–Wilson polynomial, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $z$ : 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 $p_{n}\left(x;a,b,c,d\,|\,q\right)$ are symmetric in the parameters $a, b, c, d$ .

Orthogonality

ⓘ

Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §18.28(ii), §18.28 and Ch.18

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		$2\pi\sin\theta\,w(\cos\theta)={\left\|\frac{\left({\mathrm{e}}^{2i\theta};q% \right)_{\infty}}{\left(a{\mathrm{e}}^{\mathrm{i}\theta},b{\mathrm{e}}^{% \mathrm{i}\theta},c{\mathrm{e}}^{\mathrm{i}\theta},d{\mathrm{e}}^{\mathrm{i}% \theta};q\right)_{\infty}}\right\|}^{2},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : 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), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $\sin\NVar{z}$ : sine function, $w(x)$ : weight function and $q$ : real variable Permalink: http://dlmf.nist.gov/18.28.E3 Encodings: TeX, pMML, png See also: Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

18.28.4		$h_{0}=\frac{\left(abcd;q\right)_{\infty}}{\left(q,ab,ac,ad,bc,bd,cd;q\right)_{% \infty}},$
ⓘ 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, $q$ : real variable and $h_{n}$ Permalink: http://dlmf.nist.gov/18.28.E4 Encodings: TeX, pMML, png See also: Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

18.28.5		$h_{n}=h_{0}\frac{(1-abcdq^{n-1})\left(q,ab,ac,ad,bc,bd,cd;q\right)_{n}}{(1-% abcdq^{2n-1})\left(abcd;q\right)_{n}},$
$n=1,2,\dotsc$ .
ⓘ 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, $q$ : real variable, $n$ : nonnegative integer and $h_{n}$ Permalink: http://dlmf.nist.gov/18.28.E5 Encodings: TeX, pMML, png See also: Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

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

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Keywords:: Askey–Wilson polynomials, $q$ -difference equation
Modification (effective with 1.2.0):: This paragraph as well as (18.28.6_1)–(18.28.6_2) were added.
See also:: Annotations for §18.28(ii), §18.28 and Ch.18

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

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Keywords:: Askey–Wilson polynomials, recurrence relation
Modification (effective with 1.2.0):: This paragraph as well as (18.28.6_3)–(18.28.6_4) were added.
See also:: Annotations for §18.28(ii), §18.28 and Ch.18

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	$\displaystyle a_{n}$	$\displaystyle=\frac{(1-abq^{n})(1-acq^{n})(1-adq^{n})(1-abcdq^{n-1})}{a(1-% abcdq^{2n-1})(1-abcdq^{2n})},$
18.28.6_4	$\displaystyle c_{n}$	$\displaystyle=\frac{a(1-q^{n})(1-bcq^{n-1})(1-bdq^{n-1})(1-cdq^{n-1})}{(1-% abcdq^{2n-2})(1-abcdq^{2n-1})}.$
ⓘ 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

ⓘ

Keywords:: Askey–Wilson polynomials, duality
Modification (effective with 1.2.0):: This paragraph as well as (18.28.6_5) were added.
See also:: Annotations for §18.28(ii), §18.28 and Ch.18

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

ⓘ

Notes:: See Ismail (2009, (15.1.5), (15.1.6), (15.1.11)).
Permalink:: http://dlmf.nist.gov/18.28.iii
Modification (effective with 1.2.0):: The last sentence of this subsection which originally stated “More generally, without the constraints $|a|,|b|\leq 1$ ” has been updated to be “More generally, if $|ab|\leq 1$ instead of $|a|,|b|\leq 1$ ”.
See also:: Annotations for §18.28 and Ch.18

18.28.7		$Q_{n}\left(\cos\theta;a,b\,\|\,q\right)=p_{n}\left(\cos\theta;a,b,0,0\,\|\,q% \right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}\frac{\left(abq^{\ell};q\right)_{n-\ell% }\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}\*\prod_{j=0}^{\ell-1}(% 1-2aq^{j}\cos\theta+a^{2}q^{2j})=\frac{\left(ab;q\right)_{n}}{a^{n}}{{}_{3}% \phi_{2}}\left({q^{-n},a{\mathrm{e}}^{\mathrm{i}\theta},a{\mathrm{e}}^{-% \mathrm{i}\theta}\atop ab,0};q,q\right)=\left(b{\mathrm{e}}^{-\mathrm{i}\theta% };q\right)_{n}{\mathrm{e}}^{\mathrm{i}n\theta}{{}_{2}\phi_{1}}\left({q^{-n},a{% \mathrm{e}}^{\mathrm{i}\theta}\atop b^{-1}q^{1-n}{\mathrm{e}}^{\mathrm{i}% \theta}};q,b^{-1}q{\mathrm{e}}^{-\mathrm{i}\theta}\right).$
ⓘ Defines: $Q_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\,\|\,\NVar{q}\right)$ : Al-Salam–Chihara polynomial Symbols: $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\,\|\,\NVar{q}\right)$ : Askey–Wilson polynomial, $\cos\NVar{z}$ : 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.E7 Encodings: TeX, pMML, png See also: Annotations for §18.28(iii), §18.28 and Ch.18

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 $a=\overline{b}$ ; $ab\neq 1$ ; $\|a\|,\|b\|\leq 1$ .
ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\,\|\,\NVar{q}\right)$ : Al-Salam–Chihara polynomial, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $\mathbb{R}$ : 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 originally stated that “ $\|ab\|<1$ ” has been updated to be “ $ab\neq 1$ ”. See also: Annotations for §18.28(iii), §18.28 and Ch.18

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

ⓘ

Keywords:: Al-Salam–Chihara polynomials, $q^{-1}$ -Al-Salam–Chihara polynomials
Notes:: See Askey and Ismail (1984, Chapter 3).
Referenced by:: Erratum (V1.2.0) for Subsection 18.28(iv)
Permalink:: http://dlmf.nist.gov/18.28.iv
Clarification (effective with 1.2.0):: At the end of the subsection the text which originally stated “then the measure in (18.28.10) is uniquely determined” has been updated to be “then the measure in (18.28.10) is the unique orthogonality measure”.
See also:: Annotations for §18.28 and Ch.18

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		$\sum_{y=0}^{\infty}\frac{(1-q^{2y}a^{-2})\left(a^{-2},(ab)^{-1};q\right)_{y}}{% (1-a^{-2})\left(q,bqa^{-1};q\right)_{y}}(ba^{-1})^{y}q^{y^{2}}\Q_{n}\left(% \tfrac{1}{2}(aq^{-y}+a^{-1}q^{y});a,b\,\|\,q^{-1}\right)\Q_{m}\left(\tfrac{1}{% 2}(aq^{-y}+a^{-1}q^{y});a,b\,\|\,q^{-1}\right)=\frac{\left(qa^{-2};q\right)_{% \infty}}{\left(ba^{-1}q;q\right)_{\infty}}\left(q,(ab)^{-1};q\right)_{n}(ab)^{% n}q^{-n^{2}}\delta_{n,m}.$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $Q_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\,\|\,\NVar{q}^{-1}\right)$ : $q^{-1}$ -Al-Salam–Chihara polynomial, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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		$0<q<1,a,b\in\mathbb{R},ab>1,a^{-1}b<q^{-1},$
ⓘ 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

18.28.12		$0<q<1,\ifrac{a}{\mathrm{i}},\ifrac{b}{\mathrm{i}}\in\mathbb{R},(\Im a)(\Im b)>% 0,a^{-1}b<q^{-1}.$
ⓘ 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 $q<a^{-1}b<q^{-1}$ , 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

ⓘ

Keywords:: continuous $q$ -ultraspherical polynomials
Notes:: See Gasper and Rahman (2004, (7.4.2), (7.4.14)–(7.4.16)) and Ismail (2009, (13.2.3)–(13.2.5), (13.2.11)).
Permalink:: http://dlmf.nist.gov/18.28.v
See also:: Annotations for §18.28 and Ch.18

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	$\displaystyle C_{n}\left(\cos\theta;\beta\,\|\,q\right)$	$\displaystyle=\frac{\left(\beta^{2};q\right)_{n}}{\left(q;q\right)_{n}\beta^{% \frac{1}{2}n}}{{}_{4}\phi_{3}}\left({q^{-n},\beta^{2}q^{n},\beta^{\frac{1}{2}}% {\mathrm{e}}^{\mathrm{i}\theta},\beta^{\frac{1}{2}}{\mathrm{e}}^{-\mathrm{i}% \theta}\atop\beta q^{\frac{1}{2}},-\beta,-\beta q^{\frac{1}{2}}};q,q\right).$
ⓘ 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, $C_{\NVar{n}}\left(\NVar{x};\NVar{\beta}\,\|\,\NVar{q}\right)$ : 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		$\frac{1}{2\pi}\int_{0}^{\pi}C_{n}\left(\cos\theta;\beta\,\|\,q\right)C_{m}\left% (\cos\theta;\beta\,\|\,q\right)\*{\left\|\frac{\left({\mathrm{e}}^{2\mathrm{i}% \theta};q\right)_{\infty}}{\left(\beta{\mathrm{e}}^{2\mathrm{i}\theta};q\right% )_{\infty}}\right\|}^{2}\,\mathrm{d}\theta=\frac{\left(\beta,\beta q;q\right)_{% \infty}}{\left(\beta^{2},q;q\right)_{\infty}}\frac{(1-\beta)\left(\beta^{2};q% \right)_{n}}{(1-\beta q^{n})\left(q;q\right)_{n}}\delta_{n,m},$
$-1<\beta<1$ .
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $C_{\NVar{n}}\left(\NVar{x};\NVar{\beta}\,\|\,\NVar{q}\right)$ : 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(vi) Continuous $q$ -Hermite Polynomials

ⓘ

Keywords:: continuous $q$ -Hermite polynomials
Notes:: See Ismail (2009, (13.1.7), (13.1.11)).
Permalink:: http://dlmf.nist.gov/18.28.vi
See also:: Annotations for §18.28 and Ch.18

18.28.16		$H_{n}\left(\cos\theta\,\|\,q\right)=\sum_{\ell=0}^{n}\frac{\left(q;q\right)_{n}% {\mathrm{e}}^{\mathrm{i}(n-2\ell)\theta}}{\left(q;q\right)_{\ell}\left(q;q% \right)_{n-\ell}}={\mathrm{e}}^{\mathrm{i}n\theta}{{}_{2}\phi_{0}}\left({q^{-n% },0\atop-};q,q^{n}{\mathrm{e}}^{-2\mathrm{i}\theta}\right).$
ⓘ Defines: $H_{\NVar{n}}\left(\NVar{x}\,\|\,\NVar{q}\right)$ : continuous $q$ -Hermite 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.E16 Encodings: TeX, pMML, png See also: Annotations for §18.28(vi), §18.28 and Ch.18

18.28.17		$\frac{1}{2\pi}\int_{0}^{\pi}H_{n}\left(\cos\theta\,\|\,q\right)H_{m}\left(\cos% \theta\,\|\,q\right){\left\|\left({\mathrm{e}}^{2\mathrm{i}\theta};q\right)_{% \infty}\right\|}^{2}\,\mathrm{d}\theta=\frac{\delta_{n,m}}{\left(q^{n+1};q% \right)_{\infty}}.$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\pi$ : the ratio of the circumference of a circle to its diameter, $H_{\NVar{n}}\left(\NVar{x}\,\|\,\NVar{q}\right)$ : continuous $q$ -Hermite polynomial, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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(vii) Continuous $q^{-1}$ -Hermite Polynomials

ⓘ

Keywords:: continuous $q^{-1}$ -Hermite polynomials
Notes:: See Ismail (2009, (21.2.1), (21.2.5)).
Permalink:: http://dlmf.nist.gov/18.28.vii
See also:: Annotations for §18.28 and Ch.18

18.28.18		$h_{n}\left(\sinh t\,\|\,q\right)=\sum_{\ell=0}^{n}q^{\frac{1}{2}\ell(\ell+1)}% \frac{\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}{\mathrm{e}}^{(n-2% \ell)t}={\mathrm{e}}^{nt}{{}_{1}\phi_{1}}\left({q^{-n}\atop 0};q,-q{\mathrm{e}% }^{-2t}\right)={\mathrm{i}}^{-n}H_{n}\left(\mathrm{i}\sinh t\,\|\,q^{-1}\right).$
ⓘ Defines: $h_{\NVar{n}}\left(\NVar{x}\,\|\,\NVar{q}\right)$ : continuous $q^{-1}$ -Hermite polynomial Symbols: $H_{\NVar{n}}\left(\NVar{x}\,\|\,\NVar{q}\right)$ : continuous $q$ -Hermite polynomial, $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : 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, $t$ : real variable, $q$ : real variable, $\ell$ : 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.

§18.28(viii) $q$ -Racah Polynomials

ⓘ

Keywords:: $q$ -Racah polynomials, relation to $q$ -hypergeometric function
Notes:: See Gasper and Rahman (2004, (7.2.11)) and Ismail (2009, (15.6.1), (15.6.7)).
Referenced by:: §18.28(xi), Erratum (V1.2.0) §18.28
Permalink:: http://dlmf.nist.gov/18.28.viii
Addition (effective with 1.2.0):: Instead of referring to Koekoek et al. (2010, Eq. (14.2.2)) for $\omega_{y}$ and $h_{n}$ we now explicitly give these results in Equations (18.28.21)–(18.28.23) which were added. Everything underneath these equations is new material.
See also:: Annotations for §18.28 and Ch.18

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		$\sum_{y=0}^{N}R_{n}(q^{-y}+\gamma\delta q^{y+1})R_{m}(q^{-y}+\gamma\delta q^{y% +1})\omega_{y}=h_{n}\delta_{n,m},$
$n,m=0,1,\dots,N$ ,
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : 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		$\omega_{y}=\frac{\left(\alpha q,\beta\delta q,\gamma q,\gamma\delta q;q\right)% _{y}}{\left(q,\frac{\gamma\delta}{\alpha}q,\frac{\gamma}{\beta}q,\delta q;q% \right)_{y}}\frac{1-\gamma\delta q^{2y+1}}{\left(\alpha\beta q\right)^{y}},$
ⓘ Symbols: $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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		$h_{n}=\frac{\left(\alpha\beta\right)^{n+1}q^{\left(n+1\right)^{2}}}{\alpha% \beta q^{2n+1}-1}\frac{\left(q;q\right)_{n}}{\left(\alpha q,\beta\delta q,% \gamma q;q\right)_{n}}\*\frac{\left(\frac{\gamma}{\alpha\beta}q^{-n},\frac{% \delta}{\alpha}q^{-n},\frac{1}{\beta}q^{-n},\gamma\delta q;q\right)_{\infty}}{% \left(\frac{1}{\alpha\beta}q^{-n},\frac{\gamma\delta}{\alpha}q,\frac{\gamma}{% \beta}q,\delta q;q\right)_{\infty}}.$
ⓘ 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, $\delta$ : 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

Duality

ⓘ

Addition (effective with 1.2.0):: This paragraph as well as (18.28.23) were added.
See also:: Annotations for §18.28(viii), §18.28 and Ch.18

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 $\gamma q=q^{-N}$ ; $n,y=0,1,\ldots,N$ .
ⓘ Symbols: $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\,\|\,\NVar{q}\right)$ : $q$ -Racah polynomial, $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant, $q$ : real variable and $n$ : nonnegative integer Referenced by: §18.28(viii), §18.28(viii), Erratum (V1.2.0) §18.28 Permalink: http://dlmf.nist.gov/18.28.E23 Encodings: TeX, pMML, png See also: Annotations for §18.28(viii), §18.28(viii), §18.28 and Ch.18

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

ⓘ

Referenced by:: Erratum (V1.2.0) §18.28, Version 1.2.0 (March 27, 2024)
Permalink:: http://dlmf.nist.gov/18.28.ix
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §18.28 and Ch.18

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		$\frac{\left(q;q\right)_{n}}{\left(q^{\alpha+1};q\right)_{n}}P_{n}^{(\alpha,% \beta)}(x\,\|\,q)=R_{n}\left(z;q^{\frac{1}{2}\alpha+\frac{1}{4}},q^{\frac{1}{2}% \alpha+\frac{3}{4}},-q^{\frac{1}{2}\beta+\frac{1}{4}},-q^{\frac{1}{2}\beta+% \frac{3}{4}}\,\|\,q\right)={{}_{4}\phi_{3}}\left({q^{-n},q^{n+\alpha+\beta+1},q% ^{\frac{1}{2}\alpha+\frac{1}{4}}z,q^{\frac{1}{2}\alpha+\frac{1}{4}}z^{-1}\atop q% ^{\alpha+1},-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}% ;q,q\right).$
ⓘ Symbols: $(\NVar{a},\NVar{b})$ : open interval, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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, $z$ : complex variable, $q$ : real variable, $n$ : nonnegative integer and $x$ : real variable Sources: Askey and Wilson (1985, (4.16)); Gasper and Rahman (2004, (7.5.24)) Permalink: http://dlmf.nist.gov/18.28.E24 Encodings: TeX, pMML, png See also: Annotations for §18.28(ix), §18.28 and Ch.18

Specialization to continuous $q$ -ultraspherical:

18.28.25		$P_{n}^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x\,\|\,q)=\frac{q^{\frac{1}{2% }n\lambda}\left(q^{\lambda+\ifrac{1}{2}};q\right)_{n}}{\left(q^{2\lambda};q% \right)_{n}}C_{n}\left(x;q^{\lambda}\,\|\,q\right).$
ⓘ Symbols: $(\NVar{a},\NVar{b})$ : open interval, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $C_{\NVar{n}}\left(\NVar{x};\NVar{\beta}\,\|\,\NVar{q}\right)$ : continuous $q$ -ultraspherical polynomial, $q$ : real variable, $n$ : nonnegative integer and $x$ : real variable Proved: Gasper and Rahman (2004, (7.5.33))(proved) Permalink: http://dlmf.nist.gov/18.28.E25 Encodings: TeX, pMML, png See also: Annotations for §18.28(ix), §18.28 and Ch.18

§18.28(x) Limit Relations

ⓘ

Keywords:: Askey–Wilson class orthogonal polynomials, limit relations
Permalink:: http://dlmf.nist.gov/18.28.x
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §18.28 and Ch.18

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

ⓘ

Addition (effective with 1.2.0):: This paragraph as well as (18.28.26) were added.
See also:: Annotations for §18.28(x), §18.28 and Ch.18

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

ⓘ

Addition (effective with 1.2.0):: This paragraph as well as (18.28.27)–(18.28.28) were added.
See also:: Annotations for §18.28(x), §18.28 and Ch.18

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

From Askey–Wilson to Wilson

ⓘ

Addition (effective with 1.2.0):: This paragraph as well as (18.28.29) were added.
See also:: Annotations for §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

ⓘ

Addition (effective with 1.2.0):: This paragraph as well as (18.28.30) were added.
See also:: Annotations for §18.28(x), §18.28 and Ch.18

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

ⓘ

Addition (effective with 1.2.0):: This paragraph as well as (18.28.31) were added.
See also:: Annotations for §18.28(x), §18.28 and Ch.18

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

ⓘ

Addition (effective with 1.2.0):: This paragraph as well as (18.28.32) were added.
See also:: Annotations for §18.28(x), §18.28 and Ch.18

18.28.32		$\lim_{\beta\to 0}C_{n}\left(x;\beta\,\|\,q\right)=\frac{H_{n}\left(x\,\|\,q% \right)}{\left(q;q\right)_{n}}.$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\,\|\,\NVar{q}\right)$ : continuous $q$ -Hermite polynomial, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $C_{\NVar{n}}\left(\NVar{x};\NVar{\beta}\,\|\,\NVar{q}\right)$ : continuous $q$ -ultraspherical polynomial, $q$ : real variable, $n$ : nonnegative integer and $x$ : real variable Source: Koekoek et al. (2010, (14.10.34)) Referenced by: §18.28(x) Permalink: http://dlmf.nist.gov/18.28.E32 Encodings: TeX, pMML, png See also: Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

From Continuous $q$ -Hermite to Hermite

ⓘ

Addition (effective with 1.2.0):: This paragraph as well as (18.28.33) were added.
See also:: Annotations for §18.28(x), §18.28 and Ch.18

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

ⓘ

Addition (effective with 1.2.0):: This paragraph as well as (18.28.34) were added.
See also:: Annotations for §18.28(x), §18.28 and Ch.18

18.28.34		$\lim_{q\to 1}R_{n}\left(q^{-y}+q^{y+\gamma+\delta+1};q^{\alpha},q^{\beta},q^{% \gamma},q^{\delta}\,\|\,q\right)=R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,% \gamma,\delta\right).$
ⓘ Symbols: $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\,\|\,\NVar{q}\right)$ : $q$ -Racah polynomial, $y$ : real variable, $\delta$ : arbitrary small positive constant, $q$ : real variable and $n$ : nonnegative integer Source: Koekoek et al. (2010, (14.2.24)) Referenced by: §18.28(x), Erratum (V1.2.0) §18.28 Permalink: http://dlmf.nist.gov/18.28.E34 Encodings: TeX, pMML, png See also: Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

§18.28(xi) Limits for $q\downarrow-1$

ⓘ

Keywords:: Askey–Wilson class orthogonal polynomials, Askey–Wilson polynomials, Bannai–Ito polynomials, $q$ -hypergeometric function, relation to $q$ -hypergeometric functions, relations to other functions, representation as $q$ -hypergeometric functions
Referenced by:: Erratum (V1.2.0) §18.28, Version 1.2.0 (March 27, 2024)
Permalink:: http://dlmf.nist.gov/18.28.xi
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §18.28 and Ch.18

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).

18.27

q

-Hahn Class 18.29 Asymptotic Approximations for

q

-Hahn and Askey–Wilson Classes

NIST

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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 $a=\overline{b}$ ; $ab\neq 1$ ; $\|a\|,\|b\|\leq 1$ .
ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\,\|\,\NVar{q}\right)$ : Al-Salam–Chihara polynomial, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $\mathbb{R}$ : 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 originally stated that “ $\|ab\|<1$ ” has been updated to be “ $ab\neq 1$ ”. See also: Annotations for §18.28(iii), §18.28 and Ch.18

§18.28 Askey–Wilson Class

Contents

§18.28(i) Introduction

§18.28(ii) Askey–Wilson Polynomials

Orthogonality

q-Difference Equation

Recurrence Relation

Duality

§18.28(iii) Al-Salam–Chihara Polynomials

§18.28(iv) q−1-Al-Salam–Chihara Polynomials

§18.28(v) Continuous q-Ultraspherical Polynomials

§18.28(vi) Continuous q-Hermite Polynomials

§18.28(vii) Continuous q−1-Hermite Polynomials

§18.28(viii) q-Racah Polynomials

Duality

§18.28(ix) Continuous q-Jacobi Polynomials

§18.28(x) Limit Relations

From Askey–Wilson to Big q-Jacobi

From Askey–Wilson to Little q-Jacobi

From Askey–Wilson to Wilson

From Continuous q-Jacobi to Jacobi

From Continuous q-Ultraspherical to Ultraspherical

From Continuous q-Ultraspherical to Continuous q-Hermite

From Continuous q-Hermite to Hermite

From q-Racah to Racah

§18.28(xi) Limits for q↓−1

$q$ -Difference Equation

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

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

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

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

§18.28(viii) $q$ -Racah Polynomials

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

From Askey–Wilson to Big $q$ -Jacobi

From Askey–Wilson to Little $q$ -Jacobi

From Continuous $q$ -Jacobi to Jacobi

From Continuous $q$ -Ultraspherical to Ultraspherical

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

From Continuous $q$ -Hermite to Hermite

From $q$ -Racah to Racah

§18.28(xi) Limits for $q\downarrow-1$