# §18.27(i) Introduction

The $q$-hypergeometric OP’s comprise the $q$-Hahn class OP’s and the Askey–Wilson class OP’s (§18.28). For the notation of $q$-hypergeometric functions see §§17.2 and 17.4(i).

The $q$-Hahn class OP’s comprise systems of OP’s $\{p_{n}(x)\}$, $n=0,1,\dots,N$, or $n=0,1,2,\dots$, that are eigenfunctions of a second-order $q$-difference operator. Thus

 18.27.1 $A(x)p_{n}(qx)+B(x)p_{n}(x)+C(x)p_{n}(q^{-1}x)=\lambda_{n}p_{n}(x),$

where $A(x)$, $B(x)$, and $C(x)$ are independent of $n$, and where the $\lambda_{n}$ are the eigenvalues. In the $q$-Hahn class OP’s the role of the operator $\ifrac{d}{dx}$ in the Jacobi, Laguerre, and Hermite cases is played by the $q$-derivative $\mathcal{D}_{q}$, as defined in (17.2.41). A (nonexhaustive) classification of such systems of OP’s was made by Hahn (1949). There are 18 families of OP’s of $q$-Hahn class. These families depend on further parameters, in addition to $q$. The generic (top level) cases are the $q$-Hahn polynomials and the big $q$-Jacobi polynomials, each of which depends on three further parameters.

All these systems of OP’s have orthogonality properties of the form

 18.27.2 $\sum_{x\in X}p_{n}(x)p_{m}(x)\,|x|\,v_{x}=h_{n}\delta_{n,m},$

where $X$ is given by $X=\{aq^{y}\}_{y\in I_{+}}$ or $X=\{aq^{y}\}_{y\in I_{+}}\cup\{-bq^{y}\}_{y\in I_{-}}$. Here $a,b$ are fixed positive real numbers, and $I_{+}$ and $I_{-}$ are sequences of successive integers, finite or unbounded in one direction, or unbounded in both directions. If $I_{+}$ and $I_{-}$ are both nonempty, then they are both unbounded to the right. Some of the systems of OP’s that occur in the classification do not have a unique orthogonality property. Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval.

Here only a few families are mentioned. They are defined by their $q$-hypergeometric representations, followed by their orthogonality properties. For other formulas, including $q$-difference equations, recurrence relations, duality formulas, special cases, and limit relations, see Koekoek et al. (2010, Chapter 14). See also Gasper and Rahman (2004, pp. 195–199, 228–230) and Ismail (2005, Chapters 13, 18, 21).

# §18.27(ii) $q$-Hahn Polynomials

 18.27.3 $Q_{n}(x)=\mathop{Q_{n}\/}\nolimits\!\left(x;\alpha,\beta,N;q\right)=\mathop{{{% }_{3}\phi_{2}}\/}\nolimits\!\left({q^{-n},\alpha\beta q^{n+1},x\atop\alpha q,q% ^{-N}};q,q\right),$ $n=0,1,\dots,N$. Defines: $\mathop{Q_{n}\/}\nolimits\!\left(x;\alpha,\beta,N;q\right)$: $q$-Hahn polynomial Symbols: $\mathop{{{}_{r+1}\phi_{s}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_% {1},b_{2},\dots,b_{s}};q,z\right)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: real variable, $n$: nonnegative integer, $N$: positive integer and $x$: real variable Referenced by: §18.27(ii) Permalink: http://dlmf.nist.gov/18.27.E3 Encodings: TeX, pMML, png
 18.27.4 $\sum_{y=0}^{N}Q_{n}(q^{-y})Q_{m}(q^{-y})\frac{(\alpha q,q^{-N};q)_{y}(\alpha% \beta q)^{-y}}{(q,\beta^{-1}q^{-N};q)_{y}}=h_{n}\delta_{n,m},$ $n,m=0,1,\ldots,N$.

For $h_{n}$ see Koekoek et al. (2010, Eq. (14.6.2)).

# §18.27(iii) Big $q$-Jacobi Polynomials

 18.27.5 $\mathop{P_{n}\/}\nolimits\!\left(x;a,b,c;q\right)=\mathop{{{}_{3}\phi_{2}}\/}% \nolimits\!\left({q^{-n},abq^{n+1},x\atop aq,cq};q,q\right),$ Defines: $\mathop{P_{n}\/}\nolimits\!\left(x;a,b,c;q\right)$: big $q$-Jacobi polynomial Symbols: $\mathop{{{}_{r+1}\phi_{s}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_% {1},b_{2},\dots,b_{s}};q,z\right)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: real variable, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.27.E5 Encodings: TeX, pMML, png

and

 18.27.6 $\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(x;c,d;q\right)=\frac{c^{n}q^% {-(\alpha+1)n}\left(q^{\alpha+1},-q^{\alpha+1}c^{-1}d;q\right)_{n}}{\left(q,-q% ;q\right)_{n}}\*\mathop{P_{n}\/}\nolimits\!\left(q^{\alpha+1}c^{-1}dx;q^{% \alpha},q^{\beta},-q^{\alpha}c^{-1}d;q\right).$ Defines: $\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(x;c,d;q\right)$: big $q$-Jacobi polynomial Symbols: $(a,b)$: open interval, $\mathop{P_{n}\/}\nolimits\!\left(x;a,b,c;q\right)$: big $q$-Jacobi polynomial, $\left(a;q\right)_{n}$: $q$-factorial (or $q$-shifted factorial), $q$: real variable, $n$: nonnegative integer, $x$: real variable and $P_{n}^{(\alpha,\beta)}(x;c,d;q)$ Permalink: http://dlmf.nist.gov/18.27.E6 Encodings: TeX, pMML, png

The orthogonality relations are given by (18.27.2), with

 18.27.7 $p_{n}(x)=\mathop{P_{n}\/}\nolimits\!\left(x;a,b,c;q\right),$
 18.27.8 ${X=\{aq^{\ell+1}\}_{\ell=0,1,2,\ldots}\cup\{cq^{\ell+1}\}_{\ell=0,1,2,\ldots}},$
 18.27.9 $v_{x}=\frac{(a^{-1}x,c^{-1}x;q)_{\infty}}{(x,bc^{-1}x;q)_{\infty}},$ $0, $0, $c<0$, Symbols: $q$: real variable, $x$: real variable and $v_{x}$ Permalink: http://dlmf.nist.gov/18.27.E9 Encodings: TeX, pMML, png

and

 18.27.10 $p_{n}(x)=\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(x;c,d;q\right)$
 18.27.11 $X=\{cq^{\ell}\}_{\ell=0,1,2,\ldots}\cup\{-dq^{\ell}\}_{\ell=0,1,2,\ldots},$
 18.27.12 $v_{x}=\frac{(qx/c,-qx/d;q)_{\infty}}{(q^{\alpha+1}x/c,-q^{\beta+1}x/d;q)_{% \infty}},$ $\alpha,\beta>-1$, $c,d>0$. Symbols: $q$: real variable, $x$: real variable and $v_{x}$ Permalink: http://dlmf.nist.gov/18.27.E12 Encodings: TeX, pMML, png

For $h_{n}$ see Koekoek et al. (2010, Eq. (14.5.2)).

# §18.27(iv) Little $q$-Jacobi Polynomials

 18.27.13 $p_{n}(x)=\mathop{p_{n}\/}\nolimits\!\left(x;a,b;q\right)=\mathop{{{}_{2}\phi_{% 1}}\/}\nolimits\!\left({q^{-n},abq^{n+1}\atop aq};q,qx\right).$ Defines: $\mathop{p_{n}\/}\nolimits\!\left(x;a,b;q\right)$: little $q$-Jacobi polynomial Symbols: $\mathop{{{}_{r+1}\phi_{s}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_% {1},b_{2},\dots,b_{s}};q,z\right)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: real variable, $n$: nonnegative integer, $p_{n}(x)$: polynomial of degree $n$ and $x$: real variable Permalink: http://dlmf.nist.gov/18.27.E13 Encodings: TeX, pMML, png
 18.27.14 $\sum_{y=0}^{\infty}p_{n}(q^{y})p_{m}(q^{y})\frac{\left(bq;q\right)_{y}(aq)^{y}% }{\left(q;q\right)_{y}}=h_{n}\delta_{n,m},$ $0 .

For $h_{n}$ see Koekoek et al. (2010, Eq. (14.12.2)).

Bounds for the extreme zeros are given in Driver and Jordaan (2013).

# §18.27(v) $q$-Laguerre Polynomials

 18.27.15 $\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(x;q\right)=\frac{\left(q^{\alpha+1% };q\right)_{n}}{\left(q;q\right)_{n}}\mathop{{{}_{1}\phi_{1}}\/}\nolimits\!% \left({q^{-n}\atop q^{\alpha+1}};q,-xq^{n+\alpha+1}\right).$ Defines: $\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(x;q\right)$: $q$-Laguerre 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, $q$: real variable, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.27.E15 Encodings: TeX, pMML, png

The measure is not uniquely determined:

 18.27.16 $\int_{0}^{\infty}\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(x;q\right)\mathop% {L^{(\alpha)}_{m}\/}\nolimits\!\left(x;q\right)\frac{x^{\alpha}}{\left(-x;q% \right)_{\infty}}dx=\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n% }q^{n}}h_{0}^{(1)}\delta_{n,m},$ $\alpha>-1$,

where $h_{0}^{(1)}$ is given in Koekoek et al. (2010, Eq. (14.21.2), and

 18.27.17 $\sum_{y=-\infty}^{\infty}\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(cq^{y};q% \right)\mathop{L^{(\alpha)}_{m}\/}\nolimits\!\left(cq^{y};q\right)\frac{q^{y(% \alpha+1)}}{\left(-cq^{y};q\right)_{\infty}}=\frac{\left(q^{\alpha+1};q\right)% _{n}}{\left(q;q\right)_{n}q^{n}}h_{0}^{(2)}\delta_{n,m},$ $\alpha>-1$, $c>0$,

where $h_{0}^{(2)}$ is given in Koekoek et al. (2010, Eq. (14.21.3).

Bounds for the extreme zeros are given in Driver and Jordaan (2013).

# §18.27(vi) Stieltjes–Wigert Polynomials

 18.27.18 $\mathop{S_{n}\/}\nolimits\!\left(x;q\right)=\sum_{\ell=0}^{n}\frac{q^{\ell^{2}% }(-x)^{\ell}}{\left(q;q\right)_{\ell}\left(q;q\right)_{n-\ell}}=\frac{1}{\left% (q;q\right)_{n}}\mathop{{{}_{1}\phi_{1}}\/}\nolimits\!\left({q^{-n}\atop 0};q,% -q^{n+1}x\right).$ Defines: $\mathop{S_{n}\/}\nolimits\!\left(x;q\right)$: Stieltjes–Wigert 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, $q$: real variable, $\ell$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.27.E18 Encodings: TeX, pMML, png

(Sometimes in the literature $x$ is replaced by $q^{\frac{1}{2}}x$.)

The measure is not uniquely determined:

 18.27.19 $\int_{0}^{\infty}\frac{\mathop{S_{n}\/}\nolimits\!\left(x;q\right)\mathop{S_{m% }\/}\nolimits\!\left(x;q\right)}{\left(-x,-qx^{-1};q\right)_{\infty}}dx=\frac{% \mathop{\ln\/}\nolimits\!\left(q^{-1}\right)}{q^{n}}\frac{\left(q;q\right)_{% \infty}}{\left(q;q\right)_{n}}\delta_{n,m},$

and

 18.27.20 $\int_{0}^{\infty}\mathop{S_{n}\/}\nolimits\!\left(q^{\frac{1}{2}}x;q\right)% \mathop{S_{m}\/}\nolimits\!\left(q^{\frac{1}{2}}x;q\right)\mathop{\exp\/}% \nolimits\!\left(-\frac{(\mathop{\ln\/}\nolimits x)^{2}}{2\mathop{\ln\/}% \nolimits\!\left(q^{-1}\right)}\right)dx=\frac{\sqrt{2\pi q^{-1}\mathop{\ln\/}% \nolimits\!\left(q^{-1}\right)}}{q^{n}\left(q;q\right)_{n}}\delta_{n,m}.$

# ¶ Discrete $q$-Hermite I

 18.27.21 $\mathop{h_{n}\/}\nolimits\!\left(x;q\right)=\left(q;q\right)_{n}\sum_{\ell=0}^% {\left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}q^{\ell(\ell-1)}x^{n-2\ell}}{% \left(q^{2};q^{2}\right)_{\ell}\left(q;q\right)_{n-2\ell}}=x^{n}\mathop{{{}_{2% }\phi_{0}}\/}\nolimits\!\left({q^{-n},q^{-n+1}\atop-};q^{2},x^{-2}q^{2n-1}% \right).$ Defines: $\mathop{h_{n}\/}\nolimits\!\left(x;q\right)$: discrete $q$-Hermite I polynomial Symbols: $\left\lfloor x\right\rfloor$: floor of $x$, $\left(a;q\right)_{n}$: $q$-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/18.27.E21 Encodings: TeX, pMML, png
 18.27.22 $\sum_{\ell=0}^{\infty}\left(\mathop{h_{n}\/}\nolimits\!\left(q^{\ell};q\right)% \mathop{h_{m}\/}\nolimits\!\left(q^{\ell};q\right)+\mathop{h_{n}\/}\nolimits\!% \left(-q^{\ell};q\right)\mathop{h_{m}\/}\nolimits\!\left(-q^{\ell};q\right)% \right)\*\left(q^{\ell+1},-q^{\ell+1};q\right)_{\infty}q^{\ell}=\left(q;q% \right)_{n}\left({q,-1,-q};q\right)_{\infty}q^{n(n-1)/2}\delta_{n,m}.$

# ¶ Discrete $q$-Hermite II

 18.27.23 $\mathop{\tilde{h}_{n}\/}\nolimits\!\left(x;q\right)=\left(q;q\right)_{n}\sum_{% \ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}q^{-2n\ell}q^{\ell(2% \ell+1)}x^{n-2\ell}}{\left(q^{2};q^{2}\right)_{\ell}\left(q;q\right)_{n-2\ell}% }=x^{n}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({q^{-n},q^{-n+1}\atop 0};q^% {2},-x^{-2}q^{2}\right).$ Defines: $\mathop{\tilde{h}_{n}\/}\nolimits\!\left(x;q\right)$: discrete $q$-Hermite II polynomial Symbols: $\left\lfloor x\right\rfloor$: floor of $x$, $\left(a;q\right)_{n}$: $q$-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/18.27.E23 Encodings: TeX, pMML, png
 18.27.24 $\sum_{\ell=-\infty}^{\infty}\left(\mathop{\tilde{h}_{n}\/}\nolimits\!\left(cq^% {\ell};q\right)\mathop{\tilde{h}_{m}\/}\nolimits\!\left(cq^{\ell};q\right)+% \mathop{\tilde{h}_{n}\/}\nolimits\!\left(-cq^{\ell};q\right)\mathop{\tilde{h}_% {m}\/}\nolimits\!\left(-cq^{\ell};q\right)\right)\frac{q^{\ell}}{\left(-c^{2}q% ^{2\ell};q^{2}\right)_{\infty}}=2\frac{\left(q^{2},-c^{2}q,-c^{-2}q;q^{2}% \right)_{\infty}}{\left(q,-c^{2},-c^{-2}q^{2};q^{2}\right)_{\infty}}\frac{% \left(q;q\right)_{n}}{q^{n^{2}}}\delta_{n,m},$ $c>0$.

(For discrete $q$-Hermite II polynomials the measure is not uniquely determined.)