# Β§18.27 $q$-Hahn Class

## Β§18.27(i) Introduction

The $q$-hypergeometric OPβs comprise the $q$-Hahn class (or $q$-linear lattice class) OPβs and the AskeyβWilson class (or $q$-quadratic lattice class) OPβs (Β§18.28). Together they form the $q$-Askey scheme. This scheme gives a graphical representation of all families of OPβs belonging to it together with the limit relations between them, see Koekoek et al. (2010, p.Β 414).

For the notation of $q$-hypergeometric functions see Β§Β§17.2 and 17.4(i). Unless said otherwise, we will assume that $0. For (17.4.1) with $b_{s}=q^{-N}$, $a_{0}=q^{-m}$, and $m=0,1,\ldots,N$ we will use the convention that the summation on the right-hand side ends at $n=m$.

The $q$-Hahn class OPβs comprise systems of OPβ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 OPβs the role of the operator $\ifrac{\mathrm{d}}{\mathrm{d}x}$ 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 OPβs was made by Hahn (1949). There are 18 families of OPβ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 OPβs have orthogonality properties of the form

 18.27.2 $\sum_{x\in X}p_{n}(x)p_{m}(x)\,|x|\,v_{x}=h_{n}\delta_{n,m},$ β Defines: $v_{x}$ (locally) Symbols: $\delta_{\NVar{j},\NVar{k}}$: Kronecker delta, $\in$: element of, $p_{n}(x)$: polynomial of degree $n$, $X$: subset of $\mathbb{R}$, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $h_{n}$ Referenced by: Β§18.27(i), Β§18.27(i), Β§18.27(iii) Permalink: http://dlmf.nist.gov/18.27.E2 Encodings: TeX, pMML, png See also: Annotations for Β§18.27(i), Β§18.27 and Ch.18

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. In case of unbounded sequences (18.27.2) can be rewritten as a $q$-integral, see Β§17.2(v), and more generally Gasper and Rahman (2004, (1.11.2)). Some of the systems of OPβ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, ChapterΒ 14). See also Gasper and Rahman (2004, pp.Β 195β199, 228β230) and Ismail (2009, ChaptersΒ 13, 18, 21).

## Β§18.27(ii) $q$-Hahn Polynomials

 18.27.3 $Q_{n}(x)=Q_{n}\left(x;\alpha,\beta,N;q\right)={{}_{3}\phi_{2}}\left({q^{-n},% \alpha\beta q^{n+1},x\atop\alpha q,q^{-N}};q,q\right),$ $n=0,1,\dots,N$. β Defines: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N};\NVar{q}\right)$: $q$-Hahn polynomial 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, $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: Β§18.27(ii) Permalink: http://dlmf.nist.gov/18.27.E3 Encodings: TeX, pMML, png See also: Annotations for Β§18.27(ii), Β§18.27 and Ch.18
 18.27.4 $\sum_{y=0}^{N}Q_{n}(q^{-y})Q_{m}(q^{-y})\genfrac{[}{]}{0.0pt}{}{N}{y}_{q}\frac% {\left(\alpha q;q\right)_{y}\left(\beta q;q\right)_{N-y}}{\left(\alpha q\right% )^{y}}=h_{n}\delta_{n,m},$ $n,m=0,1,\ldots,N$, β Symbols: $\delta_{\NVar{j},\NVar{k}}$: Kronecker delta, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$: $q$-binomial coefficient (or Gaussian polynomial), $y$: real variable, $N$: positive integer, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $h_{n}$ Notes: This is Ismail (2009, (18.5.2)) but the presentation has been simplified. Referenced by: Β§18.27(ii), Erratum (V1.2.0) for Equation (18.27.4) Permalink: http://dlmf.nist.gov/18.27.E4 Encodings: TeX, pMML, png Correction (effective with 1.1.2): $q$-Pochhammer symbols now link to their definition. See also: Annotations for Β§18.27(ii), Β§18.27 and Ch.18

with

 18.27.4_1 $h_{n}=\frac{\left(\alpha q\right)^{nN}}{1-\alpha\beta q^{2n+1}}\frac{\left(% \alpha\beta q^{n+1};q\right)_{N+1}\left(\beta q;q\right)_{n}}{\genfrac{[}{]}{0% .0pt}{}{N}{n}_{q}\left(\alpha q;q\right)_{n}}.$
 18.27.4_2 $\lim_{q\to 1}Q_{n}\left(q^{-x};q^{\alpha},q^{\beta},N;q\right)=Q_{n}\left(x;% \alpha,\beta,N\right).$ β Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$: Hahn polynomial, $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N};\NVar{q}\right)$: $q$-Hahn polynomial, $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: Β§18.27(ii), Erratum (V1.2.0) Β§18.27 Permalink: http://dlmf.nist.gov/18.27.E4_2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for Β§18.27(ii), Β§18.27 and Ch.18

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

 18.27.5 $P_{n}\left(x;a,b,c;q\right)={{}_{3}\phi_{2}}\left({q^{-n},abq^{n+1},x\atop aq,% cq};q,q\right).$ β Defines: $P_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c};\NVar{q}\right)$: big $q$-Jacobi polynomial 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, $q$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: (18.27.12_5), (18.27.14_3), (18.28.26) Permalink: http://dlmf.nist.gov/18.27.E5 Encodings: TeX, pMML, png See also: Annotations for Β§18.27(iii), Β§18.27 and Ch.18

Alternative definitions and notations are

 18.27.6 $P^{(\alpha,\beta)}_{n}\left(x;c,d;q\right)=\frac{c^{n}q^{-(\alpha+1)n}\left(q^% {\alpha+1},-q^{\alpha+1}c^{-1}d;q\right)_{n}}{\left(q,-q;q\right)_{n}}\*P_{n}% \left(q^{\alpha+1}c^{-1}x;q^{\alpha},q^{\beta},-q^{\alpha}c^{-1}d;q\right),$ β Defines: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x};\NVar{c},\NVar{d};% \NVar{q}\right)$: big $q$-Jacobi polynomial and $P_{n}^{(\alpha,\beta)}(x;c,d;q)$ (locally) Symbols: $(\NVar{a},\NVar{b})$: open interval, $P_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c};\NVar{q}\right)$: big $q$-Jacobi polynomial, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: multiple $q$-Pochhammer symbol, $q$: real variable, $n$: nonnegative integer and $x$: real variable Source: Andrews and Askey (1985, (3.28)) Referenced by: (18.27.12_5), Β§18.27(iii), Erratum (V1.0.17) for Equation (18.27.6) Permalink: http://dlmf.nist.gov/18.27.E6 Encodings: TeX, pMML, png Errata (effective with 1.0.17): Originally the first argument of the big $q$-Jacobi polynomial on the right-hand side was written incorrectly as $q^{\alpha+1}c^{-1}dx$. Reported 2017-09-27 by Tom Koornwinder See also: Annotations for Β§18.27(iii), Β§18.27 and Ch.18

and

 18.27.6_5 $P_{n}(x;a,b,c,d;q)=P_{n}\left(qac^{-1}x;a,b,-ac^{-1}d;q\right).$ β Symbols: $P_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c};\NVar{q}\right)$: big $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer, $x$: real variable and $P_{n}^{(\alpha,\beta)}(x;c,d;q)$ Source: Koekoek et al. (2010, p.Β 442) Referenced by: Β§18.27(iii), Erratum (V1.2.0) Β§18.27 Permalink: http://dlmf.nist.gov/18.27.E6_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for Β§18.27(iii), Β§18.27 and Ch.18

The orthogonality relations are given by (18.27.2), with

 18.27.7 $p_{n}(x)=P_{n}\left(x;a,b,c;q\right),$
 18.27.8 ${X=\{aq^{\ell+1}\}_{\ell=0,1,2,\ldots}\cup\{cq^{\ell+1}\}_{\ell=0,1,2,\ldots}},$ β Symbols: $\cup$: union, $X$: subset of $\mathbb{R}$, $q$: real variable and $\ell$: nonnegative integer Permalink: http://dlmf.nist.gov/18.27.E8 Encodings: TeX, pMML, png See also: Annotations for Β§18.27(iii), Β§18.27 and Ch.18
 18.27.9 $v_{x}=\frac{\left(a^{-1}x,c^{-1}x;q\right)_{\infty}}{\left(x,bc^{-1}x;q\right)% _{\infty}},$ $0, $0, $c<0$, β Symbols: $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: multiple $q$-Pochhammer symbol, $q$: real variable, $x$: real variable and $v_{x}$ Permalink: http://dlmf.nist.gov/18.27.E9 Encodings: TeX, pMML, png Correction (effective with 1.1.2): $q$-Pochhammer symbols now link to their definition. See also: Annotations for Β§18.27(iii), Β§18.27 and Ch.18
 18.27.9_5 $h_{n}=\frac{\left(-c\right)^{n}a^{n+1}}{1-abq^{2n+1}}\frac{\left(q;q\right)_{n% }q^{\genfrac{(}{)}{0.0pt}{}{n+2}{2}}}{\left(aq,cq;q\right)_{n}}\*\frac{\left(q% ,c^{-1}aq,a^{-1}c,abq^{n+1};q\right)_{\infty}}{\left(aq,cq,bq^{n+1},c^{-1}abq^% {n+1};q\right)_{\infty}},$

and

 18.27.10 $p_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x;c,d;q\right)$
 18.27.11 $X=\{cq^{\ell}\}_{\ell=0,1,2,\ldots}\cup\{-dq^{\ell}\}_{\ell=0,1,2,\ldots},$ β Symbols: $\cup$: union, $X$: subset of $\mathbb{R}$, $q$: real variable and $\ell$: nonnegative integer Permalink: http://dlmf.nist.gov/18.27.E11 Encodings: TeX, pMML, png See also: Annotations for Β§18.27(iii), Β§18.27 and Ch.18
 18.27.12 $v_{x}=\frac{\left(qx/c,-qx/d;q\right)_{\infty}}{\left(q^{\alpha+1}x/c,-q^{% \beta+1}x/d;q\right)_{\infty}},$ $\alpha,\beta>-1$, $c,d>0$. β Symbols: $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: multiple $q$-Pochhammer symbol, $q$: real variable, $x$: real variable and $v_{x}$ Permalink: http://dlmf.nist.gov/18.27.E12 Encodings: TeX, pMML, png Correction (effective with 1.1.2): $q$-Pochhammer symbols now link to their definition. See also: Annotations for Β§18.27(iii), Β§18.27 and Ch.18

### From Big $q$-Jacobi to Jacobi

 18.27.12_5 $\lim_{q\to 1}P^{(\alpha,\beta)}_{n}\left(x;c,d;q\right)=\left(\frac{c+d}{2}% \right)^{n}P^{(\alpha,\beta)}_{n}\left(\frac{2x-c+d}{c+d}\right).$ β Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$: $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gaussβ hypergeometric function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x};\NVar{c},\NVar{d};% \NVar{q}\right)$: big $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable Source: Andrews and Askey (1985, (3.33)); for $c=d=1$ Proof sketch: On the left substitute first (18.27.6) and next (18.27.5). Then, after taking the limit, a ${{}_{2}F_{1}}$ hypergeometric function occurs. Finally use (18.5.7). Referenced by: Β§18.27(iii), Β§18.27(iii), Erratum (V1.2.0) Β§18.27 Permalink: http://dlmf.nist.gov/18.27.E12_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for Β§18.27(iii), Β§18.27(iii), Β§18.27 and Ch.18

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

 18.27.13 $p_{n}(x)=p_{n}\left(x;a,b;q\right)={{}_{2}\phi_{1}}\left({q^{-n},abq^{n+1}% \atop aq};q,qx\right)=(-b)^{-n}q^{\ifrac{-n(n+1)}{2}}\frac{\left(qb;q\right)_{% n}}{\left(qa;q\right)_{n}}\*{{}_{3}\phi_{2}}\left({q^{-n},abq^{n+1},qbx\atop qb% ,0};q,q\right).$ β Defines: $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b};\NVar{q}\right)$: little $q$-Jacobi 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, $p_{n}(x)$: polynomial of degree $n$, $q$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: (18.28.27), (18.28.28), Erratum (V1.2.0) for Equation (18.27.13) Permalink: http://dlmf.nist.gov/18.27.E13 Encodings: TeX, pMML, png Addition (effective with 1.2.0): The ${{}_{3}\phi_{2}}$ representation was added. See also: Annotations for Β§18.27(iv), Β§18.27 and Ch.18
 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,

with

 18.27.14_1 $h_{n}=\frac{\left(aq\right)^{n}}{1-abq^{2n+1}}\frac{\left(q,bq;q\right)_{n}}{% \left(aq;q\right)_{n}}\*\frac{\left(abq^{n+1};q\right)_{\infty}}{\left(aq;q% \right)_{\infty}}.$ β 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 $h_{n}$ Source: Koekoek et al. (2010, (14.12.2)) Referenced by: Β§18.27(iv), (18.28.27), Erratum (V1.2.0) Β§18.27 Permalink: http://dlmf.nist.gov/18.27.E14_1 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for Β§18.27(iv), Β§18.27 and Ch.18

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

### From Big $q$-Jacobi to Little $q$-Jacobi

 18.27.14_2 $\lim_{c\to-\infty}P_{n}\left(cqx;a,b,c;q\right)=p_{n}\left(x;a,b;q\right).$
 18.27.14_3 $\lim_{c\uparrow 0}P_{n}\left(bqx;b,a,c;q\right)=(-b)^{n}q^{\ifrac{n(n+1)}{2}}% \frac{\left(qa;q\right)_{n}}{\left(qb;q\right)_{n}}p_{n}\left(x;a,b;q\right).$

### From Little $q$-Jacobi to Jacobi

 18.27.14_4 $\lim_{q\to 1}p_{n}\left(x;q^{\alpha},q^{\beta};q\right)=\frac{n!}{{\left(% \alpha+1\right)_{n}}}P^{(\alpha,\beta)}_{n}\left(1-2x\right).$

### Little $q$-Laguerre polynomials

Little $q$-Jacobi polynomials $p_{n}\left(x;a,b;q\right)$ for $b=0$ are called little $q$-Laguerre or Wall polynomials:

 18.27.14_5 $p_{n}\left(x;a,0;q\right)={{}_{2}\phi_{1}}\left({q^{-n},0\atop aq};q,qx\right).$

### From Little $q$-Laguerre to Laguerre

 18.27.14_6 $\lim_{q\to 1}p_{n}\left((1-q)x;q^{\alpha},0;q\right)=\frac{n!}{{\left(\alpha+1% \right)_{n}}}L^{(\alpha)}_{n}\left(x\right).$ β Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammerβs symbol (or shifted factorial), $!$: factorial (as in $n!$), $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b};\NVar{q}\right)$: little $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable Source: Koekoek et al. (2010, (14.12.12), (14.20.12)) Referenced by: Β§18.27(iv), Β§18.27(iv), Erratum (V1.2.0) Β§18.27 Permalink: http://dlmf.nist.gov/18.27.E14_6 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for Β§18.27(iv), Β§18.27(iv), Β§18.27 and Ch.18

## Β§18.27(v) $q$-Laguerre Polynomials

 18.27.15 $L^{(\alpha)}_{n}\left(x;q\right)=\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(% q;q\right)_{n}}{{}_{1}\phi_{1}}\left({q^{-n}\atop q^{\alpha+1}};q,-xq^{n+% \alpha+1}\right).$ β Defines: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$: $q$-Laguerre 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, $q$: real variable, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.27.E15 Encodings: TeX, pMML, png See also: Annotations for Β§18.27(v), Β§18.27 and Ch.18

The measure is not uniquely determined:

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

and

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

with

 18.27.17_1 $h_{0}^{(1)}=\frac{\left(q^{-\alpha};q\right)_{\infty}}{\left(q;q\right)_{% \infty}}\Gamma\left(\alpha+1\right)\Gamma\left(-\alpha\right),$ β Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable and $h_{n}$ Source: Koekoek et al. (2010, (14.21.2)) Referenced by: Β§18.27(v), Erratum (V1.2.0) Β§18.27 Permalink: http://dlmf.nist.gov/18.27.E17_1 Encodings: TeX, pMML, png Modification (effective with 1.2.0): This equation was added. See also: Annotations for Β§18.27(v), Β§18.27 and Ch.18
 18.27.17_2 $h_{0}^{(2)}=\frac{\left(q,-cq^{\alpha+1},-c^{-1}q^{-\alpha};q\right)_{\infty}}% {\left(q^{\alpha+1},-c,-c^{-1}q;q\right)_{\infty}}.$ β Symbols: $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: multiple $q$-Pochhammer symbol, $q$: real variable and $h_{n}$ Source: Koekoek et al. (2010, (14.21.3)) Permalink: http://dlmf.nist.gov/18.27.E17_2 Encodings: TeX, pMML, png Modification (effective with 1.2.0): This equation was added. See also: Annotations for Β§18.27(v), Β§18.27 and Ch.18

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

### From $q$-Laguerre to Laguerre

 18.27.17_3 $\lim_{q\to 1}L^{(\alpha)}_{n}\left((1-q)x;q\right)=L^{(\alpha)}_{n}\left(x% \right).$ β Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$: $q$-Laguerre polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable Source: Koekoek et al. (2010, (14.21.18)) Referenced by: Β§18.27(v), Erratum (V1.2.0) Β§18.27 Permalink: http://dlmf.nist.gov/18.27.E17_3 Encodings: TeX, pMML, png Modification (effective with 1.2.0): This equation was added. See also: Annotations for Β§18.27(v), Β§18.27(v), Β§18.27 and Ch.18

## Β§18.27(vi) StieltjesβWigert Polynomials

 18.27.18 $S_{n}\left(x;q\right)=\sum_{\ell=0}^{n}\frac{q^{\ell^{2}}(-x)^{\ell}}{\left(q;% q\right)_{\ell}\left(q;q\right)_{n-\ell}}=\frac{1}{\left(q;q\right)_{n}}{{}_{1% }\phi_{1}}\left({q^{-n}\atop 0};q,-q^{n+1}x\right).$

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

The measure is not uniquely determined:

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

and

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

### From StieltjesβWigert to Hermite

 18.27.20_5 $\lim_{q\to 1}\frac{\left(q;q\right)_{n}S_{n}\left(q^{-1}x\sqrt{2(1-q)}+1;q% \right)}{\left(\frac{1-q}{2}\right)^{n/2}}=(-1)^{n}H_{n}\left(x\right).$ β Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $S_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$: StieltjesβWigert 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 Source: Koekoek et al. (2010, (14.27.14)) Referenced by: Β§18.27(vi), Erratum (V1.2.0) Β§18.27 Permalink: http://dlmf.nist.gov/18.27.E20_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for Β§18.27(vi), Β§18.27(vi), Β§18.27 and Ch.18

## Β§18.27(vii) Discrete $q$-Hermite I and II Polynomials

### Discrete $q$-Hermite I

 18.27.21 $h_{n}\left(x;q\right)=\left(q;q\right)_{n}\sum_{\ell=0}^{\left\lfloor n/2% \right\rfloor}\frac{(-1)^{\ell}q^{\ell(\ell-1)}x^{n-2\ell}}{\left(q^{2};q^{2}% \right)_{\ell}\left(q;q\right)_{n-2\ell}}=x^{n}{{}_{2}\phi_{0}}\left({q^{-n},q% ^{-n+1}\atop-};q^{2},x^{-2}q^{2n-1}\right).$
 18.27.22 $\sum_{\ell=0}^{\infty}\left(h_{n}\left(q^{\ell};q\right)h_{m}\left(q^{\ell};q% \right)+h_{n}\left(-q^{\ell};q\right)h_{m}\left(-q^{\ell};q\right)\right)\*% \left(q^{\ell+1},-q^{\ell+1};q\right)_{\infty}q^{\ell}=\left(q;q\right)_{n}% \left({q,-1,-q};q\right)_{\infty}q^{n(n-1)/2}\delta_{n,m}.$

### Discrete $q$-Hermite II

 18.27.23 $\tilde{h}_{n}\left(x;q\right)=\left(q;q\right)_{n}\sum_{\ell=0}^{\left\lfloor n% /2\right\rfloor}\frac{(-1)^{\ell}q^{-2n\ell}q^{\ell(2\ell+1)}x^{n-2\ell}}{% \left(q^{2};q^{2}\right)_{\ell}\left(q;q\right)_{n-2\ell}}=x^{n}{{}_{2}\phi_{1% }}\left({q^{-n},q^{-n+1}\atop 0};q^{2},-x^{-2}q^{2}\right).$ β Defines: $\tilde{h}_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$: discrete $q$-Hermite II polynomial Symbols: $\left\lfloor\NVar{x}\right\rfloor$: floor of $x$, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-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/18.27.E23 Encodings: TeX, pMML, png See also: Annotations for Β§18.27(vii), Β§18.27(vii), Β§18.27 and Ch.18
 18.27.24 $\sum_{\ell=-\infty}^{\infty}\left(\tilde{h}_{n}\left(cq^{\ell};q\right)\tilde{% h}_{m}\left(cq^{\ell};q\right)+\tilde{h}_{n}\left(-cq^{\ell};q\right)\tilde{h}% _{m}\left(-cq^{\ell};q\right)\right)\frac{q^{\ell}}{\left(-c^{2}q^{2\ell};q^{2% }\right)_{\infty}}=2\frac{\left(q^{2},-c^{2}q,-c^{-2}q;q^{2}\right)_{\infty}}{% \left(q,-c^{2},-c^{-2}q^{2};q^{2}\right)_{\infty}}\frac{\left(q;q\right)_{n}}{% q^{n^{2}}}\delta_{n,m},$ $c>0$.

For discrete $q$-HermiteΒ II polynomials the measure is not uniquely determined.

### Limit Relations

 18.27.25 $\lim_{q\to 1}\frac{h_{n}\left((1-q^{2})^{\frac{1}{2}}x;q\right)}{\left(1-q^{2}% \right)^{\ifrac{n}{2}}}=2^{-n}H_{n}\left(x\right).$ β Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $h_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$: discrete $q$-Hermite I polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable Source: Koekoek et al. (2010, (14.28.12)) Referenced by: Β§18.27(vii), Β§18.27(vii), Erratum (V1.2.0) Β§18.27 Permalink: http://dlmf.nist.gov/18.27.E25 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for Β§18.27(vii), Β§18.27(vii), Β§18.27 and Ch.18
 18.27.26 $\lim_{q\to 1}\frac{\tilde{h}_{n}\left((1-q^{2})^{\frac{1}{2}}x;q\right)}{\left% (1-q^{2}\right)^{\ifrac{n}{2}}}=2^{-n}H_{n}\left(x\right).$ β Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $\tilde{h}_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$: discrete $q$-Hermite II polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable Source: Koekoek et al. (2010, (14.29.14)) Referenced by: Β§18.27(vii), Β§18.27(vii), Erratum (V1.2.0) Β§18.27 Permalink: http://dlmf.nist.gov/18.27.E26 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for Β§18.27(vii), Β§18.27(vii), Β§18.27 and Ch.18