18 Orthogonal Polynomials Other Orthogonal Polynomials 18.27

-Hahn Class 18.29 Asymptotic Approximations for

-Hahn and Askey–Wilson Classes

§18.28 Askey–Wilson Class

Keywords:: Askey–Wilson class orthogonal polynomials
Referenced by:: §18.27(i)
Permalink:: http://dlmf.nist.gov/18.28

§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 -Ultraspherical Polynomials
§18.28(vi) Continuous -Hermite Polynomials
§18.28(vii) Continuous $q^{{-1}}$ -Hermite Polynomials
§18.28(viii) -Racah Polynomials

§18.28(i) Introduction

Keywords:: Askey–Wilson class orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.28.i

The Askey–Wilson class OP’s comprise the four-parameter families of Askey–Wilson polynomials and of -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 -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 a constant. Both the Askey–Wilson polynomials and the -Racah polynomials can best be described as functions of (resp. ) 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 -Racah case, and both are eigenfunctions of a second-order -difference operator similar to (18.27.1).

In the remainder of this section the Askey–Wilson class OP’s are defined by their -hypergeometric representations, followed by their orthogonal properties. For further properties see Koekoek et al. (2010, Chapter 3). See also Gasper and Rahman (2004, pp. 180–199) and Ismail (2005, Chapter 15). For the notation of -hypergeometric functions see §§17.2 and 17.4(i).

§18.28(ii) Askey–Wilson Polynomials

Notes:: See Askey and Wilson (1985), Gasper and Rahman (2004, (7.5.2), (7.5.15), (7.5.21)), Ismail (2005, (15.2.4), (15.2.5)).
Keywords:: Askey–Wilson class orthogonal polynomials, Askey–Wilson polynomials, -hypergeometric function
Permalink:: http://dlmf.nist.gov/18.28.ii

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

Defines:: $\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,c,d\,|\,q\right)$ : Askey–Wilson polynomial
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\left(a;q\right)_{{n}}$ : -factorial (or -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 -hypergeometric) function, : real variable, $\ell$ : nonnegative integer, : nonnegative integer, $p_{n}(x)$ : polynomial of degree and : real variable
Permalink:: http://dlmf.nist.gov/18.28.E1
Encodings:: TeX, pMML, png

Assume 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, , , , , , . Then

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

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, : differential of , $\int$ : integral, : weight function, : nonnegative integer, : nonnegative integer, $p_{n}(x)$ : polynomial of degree , : real variable and $h_{n}$
Referenced by:: §18.28(ii)
Permalink:: http://dlmf.nist.gov/18.28.E2
Encodings:: TeX, pMML, png

where

18.28.3 $2\pi\mathop{\sin\/}\nolimits\theta\,w(\mathop{\cos\/}\nolimits\theta)=\left|% \frac{\left(e^{{2i\theta}};q\right)_{{\infty}}}{\left(ae^{{i\theta}},be^{{i% \theta}},ce^{{i\theta}},de^{{i\theta}};q\right)_{{\infty}}}\right|^{2},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), $\mathop{\sin\/}\nolimits z$ : sine function, : weight function and : real variable
Permalink:: http://dlmf.nist.gov/18.28.E3
Encodings:: TeX, pMML, png

18.28.4 $h_{0}=\frac{\left(abcd;q\right)_{{\infty}}}{\left(q,ab,ac,ad,bc,bd,cd;q\right)% _{{\infty}}},$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real variable and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.28.E4
Encodings:: TeX, pMML, png

18.28.5 $h_{n}=h_{0}\frac{(1-abcdq^{{n-1}})\left(q,ab,ac,ad,bc,bd,cd;q\right)_{{n}}}{(1% -abcdq^{{2n-1}})\left(abcd;q\right)_{{n}}},$ $n=1,2,\ldots$ .

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real variable, : nonnegative integer and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.28.E5
Encodings:: TeX, pMML, png

More generally, without the constraints in (18.28.2),

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

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, : differential of , $\int$ : integral, : weight function, $\ell$ : nonnegative integer, : nonnegative integer, : nonnegative integer, $p_{n}(x)$ : polynomial of degree , : real variable and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.28.E6
Encodings:: TeX, pMML, png

with 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 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. (3.1.3)) for the value of $\omega_{\ell}$ when $\alpha=a$ .

§18.28(iii) Al-Salam–Chihara Polynomials

Notes:: See Ismail (2005, (15.1.5), (15.1.6), (15.1.11)).
Permalink:: http://dlmf.nist.gov/18.28.iii

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

Defines:: $\mathop{Q_{{n}}\/}\nolimits\!\left(x;a,b\,|\,q\right)$ : Al-Salam–Chihara polynomial
Symbols:: $\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,c,d\,|\,q\right)$ : Askey–Wilson polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\left(a;q\right)_{{n}}$ : -factorial (or -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 -hypergeometric) function, : real variable, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.28.E7
Encodings:: TeX, pMML, png

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 $a=\conj{b}$ ;

; $|a|,|b|\leq 1$ .

Symbols:: $\mathop{Q_{{n}}\/}\nolimits\!\left(x;a,b\,|\,q\right)$ : Al-Salam–Chihara polynomial, $\delta_{{j,k}}$ : Kronecker delta, $\conj{z}$ : complex conjugate, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\in$ : element of, : base of exponential function, $\int$ : integral, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), $\Real$ : real line (excluding infinity), : real variable, : nonnegative integer and : nonnegative integer
Referenced by:: §18.28(iii)
Permalink:: http://dlmf.nist.gov/18.28.E8
Encodings:: TeX, pMML, png

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. (3.8.3)).

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

Notes:: See Askey and Ismail (1984, Chapter 3).
Keywords:: Al-Salam–Chihara polynomials, $q^{{-1}}$ -Al-Salam–Chihara polynomials
Permalink:: http://dlmf.nist.gov/18.28.iv

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:: $\left(a;q\right)_{{n}}$ : -factorial (or -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 -hypergeometric) function, : real variable, : real variable, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.28.E9
Encodings:: TeX, pMML, png

18.28.10 $\sum_{{y=0}}^{\infty}\frac{(1-q^{{2y}}a^{{-2}})\left(a^{{-2}},(ab)^{{-1}};q% \right)_{{y}}}{(1-a^{{-2}})\left(q,bqa^{{-1}};q\right)_{{y}}}(ba^{{-1}})^{y}q^% {{y^{2}}}\*\mathop{Q_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}(aq^{{-y}}+a^{{-1}}q% ^{y});a,b\,|\,q^{{-1}}\right)\*\mathop{Q_{{m}}\/}\nolimits\!\left(\tfrac{1}{2}% (aq^{{-y}}+a^{{-1}}q^{y});a,b\,|\,q^{{-1}}\right)=\frac{\left(qa^{{-2}};q% \right)_{{\infty}}}{\left(ba^{{-1}}q;q\right)_{{\infty}}}\left(q,(ab)^{{-1}};q% \right)_{{n}}(ab)^{n}q^{{-n^{2}}}\delta_{{n,m}}.$

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), $\mathop{Q_{{n}}\/}\nolimits\!\left(x;a,b\,|\,q^{{-1}}\right)$ : $q^{{-1}}$ -Al-Salam–Chihara polynomial, : real variable, : real variable, : nonnegative integer and : nonnegative integer
Referenced by:: §18.28(iv)
Permalink:: http://dlmf.nist.gov/18.28.E10
Encodings:: TeX, pMML, png

Eq. (18.28.10) is valid when either

18.28.11 $0<q<1,a,b\in\Real,ab>1,a^{{-1}}b<q^{{-1}},$

Symbols:: $\in$ : element of, $\Real$ : real line (excluding infinity) and : real variable
Referenced by:: §18.28(iv)
Permalink:: http://dlmf.nist.gov/18.28.E11
Encodings:: TeX, pMML, png

18.28.12 $0<q<1,\ifrac{a}{i},\ifrac{b}{i}\in\Real,(\imagpart{a})(\imagpart{b})>0,a^{{-1}% }b<q^{{-1}}.$

Symbols:: $\in$ : element of, $\imagpart{}$ : imaginary part, $\Real$ : real line (excluding infinity) and : real variable
Referenced by:: §18.28(iv)
Permalink:: http://dlmf.nist.gov/18.28.E12
Encodings:: TeX, pMML, png

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 $q<a^{{-1}}b<q^{{-1}}$ , then (18.28.10) holds with interchanged. For further nondegenerate cases see Chihara and Ismail (1993) and Christiansen and Ismail (2006).

§18.28(v) Continuous -Ultraspherical Polynomials

Notes:: See Gasper and Rahman (2004, (7.4.2), (7.4.14)–(7.4.16)) and Ismail (2005, (13.2.3)–(13.2.5), (13.2.11)).
Keywords:: continuous -ultraspherical polynomials
Permalink:: http://dlmf.nist.gov/18.28.v

18.28.13 $\mathop{C_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta;\beta\,|\,q% \right)=\sum_{{\ell=0}}^{n}\frac{\left(\beta;q\right)_{{\ell}}\left(\beta;q% \right)_{{n-\ell}}}{\left(q;q\right)_{{\ell}}\left(q;q\right)_{{n-\ell}}}e^{{i% (n-2\ell)\theta}}=\frac{\left(\beta;q\right)_{{n}}}{\left(q;q\right)_{{n}}}e^{% {in\theta}}\mathop{{{}_{{2}}\phi_{{1}}}\/}\nolimits\!\left({q^{{-n}},\beta% \atop\beta^{{-1}}q^{{1-n}}};q,\beta^{{-1}}qe^{{-2i\theta}}\right).$

Defines:: $\mathop{C_{{n}}\/}\nolimits\!\left(x;\beta\,|\,q\right)$ : continuous -ultraspherical polynomial
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\left(a;q\right)_{{n}}$ : -factorial (or -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 -hypergeometric) function, : real variable, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.28.E13
Encodings:: TeX, pMML, png

18.28.14 $\mathop{C_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta;\beta\,|\,q% \right)=\frac{\left(\beta^{2};q\right)_{{n}}}{\left(q;q\right)_{{n}}\beta^{{% \frac{1}{2}n}}}\mathop{{{}_{{4}}\phi_{{3}}}\/}\nolimits\!\left({q^{{-n}},\beta% ^{2}q^{n},\beta^{{\frac{1}{2}}}e^{{i\theta}},\beta^{{\frac{1}{2}}}e^{{-i\theta% }}\atop\beta q^{{\frac{1}{2}}},-\beta,-\beta q^{{\frac{1}{2}}}};q,q\right).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\left(a;q\right)_{{n}}$ : -factorial (or -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 -hypergeometric) function, $\mathop{C_{{n}}\/}\nolimits\!\left(x;\beta\,|\,q\right)$ : continuous -ultraspherical polynomial, : real variable and : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.28.E14
Encodings:: TeX, pMML, png

18.28.15 $\frac{1}{2\pi}\int_{0}^{\pi}\mathop{C_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta;\beta\,|\,q\right)\mathop{C_{{m}}\/}\nolimits\!\left(\mathop{% \cos\/}\nolimits\theta;\beta\,|\,q\right)\*\left|\frac{\left(e^{{2i\theta}};q% \right)_{{\infty}}}{\left(\beta e^{{2i\theta}};q\right)_{{\infty}}}\right|^{2}% d\theta=\frac{\left(\beta,\beta q;q\right)_{{\infty}}}{\left(\beta^{2},q;q% \right)_{{\infty}}}\frac{(1-\beta)\left(\beta^{2};q\right)_{{n}}}{(1-\beta q^{% n})\left(q;q\right)_{{n}}}\delta_{{n,m}},$ $-1<\beta<1$ .

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , : base of exponential function, $\int$ : integral, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), $\mathop{C_{{n}}\/}\nolimits\!\left(x;\beta\,|\,q\right)$ : continuous -ultraspherical polynomial, : real variable, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.28.E15
Encodings:: TeX, pMML, png

These polynomials are also called Rogers polynomials.

§18.28(vi) Continuous -Hermite Polynomials

Notes:: See Ismail (2005, (13.1.7), (13.1.11)).
Keywords:: continuous -Hermite polynomials
Permalink:: http://dlmf.nist.gov/18.28.vi

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 -Hermite polynomial
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\left(a;q\right)_{{n}}$ : -factorial (or -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 -hypergeometric) function, : real variable, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.28.E16
Encodings:: TeX, pMML, png

18.28.17 $\frac{1}{2\pi}\int_{0}^{\pi}\mathop{H_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\,|\,q\right)\mathop{H_{{m}}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\,|\,q\right)\left|\left(e^{{2i\theta}};q\right)_{{\infty}}% \right|^{2}d\theta=\frac{\delta_{{n,m}}}{\left(q^{{n+1}};q\right)_{{\infty}}}.$

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, $\mathop{H_{{n}}\/}\nolimits\!\left(x\,|\,q\right)$ : continuous -Hermite polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , : base of exponential function, $\int$ : integral, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real variable, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.28.E17
Encodings:: TeX, pMML, png

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

Notes:: See Ismail (2005, (21.2.1), (21.2.5)).
Keywords:: continuous $q^{{-1}}$ -Hermite polynomials
Permalink:: http://dlmf.nist.gov/18.28.vii

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

Defines:: $\mathop{h_{{n}}\/}\nolimits\!\left(x\,|\,q\right)$ : continuous $q^{{-1}}$ -Hermite polynomial
Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\,|\,q\right)$ : continuous -Hermite polynomial, : base of exponential function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\left(a;q\right)_{{n}}$ : -factorial (or -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 -hypergeometric) function, : real variable, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.28.E18
Encodings:: TeX, pMML, png

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) -Racah Polynomials

Notes:: See Gasper and Rahman (2004, (7.2.11)) and Ismail (2005, (15.6.1), (15.6.7)).
Keywords:: Askey–Wilson class orthogonal polynomials, Askey–Wilson polynomials, -Racah polynomials, -hypergeometric function
Permalink:: http://dlmf.nist.gov/18.28.viii

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:: $\mathop{R_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)$ : -Racah polynomial
Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -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 -hypergeometric) function, : real variable, : real variable, $\ell$ : nonnegative integer, : nonnegative integer, : positive integer, $\delta$ : arbitary small positive constant, : real variable and $R_{n}(x)$
Permalink:: http://dlmf.nist.gov/18.28.E19
Encodings:: TeX, pMML, png

18.28.20 $\sum_{{y=0}}^{N}R_{n}(q^{{-y}}+\gamma\delta q^{{y+1}})R_{m}(q^{{-y}}+\gamma% \delta q^{{y+1}})\omega_{y}=h_{n}\delta_{{n,m}},$ $n,m=0,1,\dots,N$ .