§18.27 $q$-Hahn Class

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

For the notation of $q$-hypergeometric functions see §§17.2 and 17.4(i). Unless said otherwise, we will assume that . For (17.4.1) with $b_s=q^{-N}$, $a_0=q^{-m}$, and $m=0,1,\mathrm{\dots },N$ we will use the convention that the summation on the right-hand side ends at $n=m$.

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

18.27.1		$$A(x)p_n(qx)+B(x)p_n(x)+C(x)p_n(q^{-1}x)={\lambda }_n{p}_n(x),$$
ⓘ Symbols: $p_n(x)$: polynomial of degree $n$, $q$: real variable, $n$: nonnegative integer, $x$: real variable, $A(x)$: coefficient, $C(x)$: coefficient, $C(x)$: coefficient and ${\lambda }_n$: eigenvalue Referenced by: §18.28(i) Permalink: http://dlmf.nist.gov/18.27.E1 Encodings: TeX, pMML, png See also: Annotations for §18.27(i), §18.27 and Ch.18

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 $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the $q$-derivative ${𝒟}_q$, 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		$$\sum _{x\in X}{p}_n(x)p_m(x)|x|v_x=h_n{\delta }_{n,m},$$
ⓘ Defines: $v_x$ (locally) Symbols: ${\delta }_{j,k}$: Kronecker delta, $\in$: element of, $p_n(x)$: polynomial of degree $n$, $X$: subset of $ℝ$, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $h_n$ Referenced by: §18.27(i), §18.27(i), §18.27(iii) Permalink: http://dlmf.nist.gov/18.27.E2 Encodings: TeX, pMML, png See also: Annotations for §18.27(i), §18.27 and Ch.18

where $X$ is given by $X={\{a{q}^y\}}_{y\in I_{+}}$ or $X={\{a{q}^y\}}_{y\in I_{+}}\cup {\{-b{q}^y\}}_{y\in I_{-}}$. 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 §17.2(v), and more generally Gasper and Rahman (2004, (1.11.2)). 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 (2009, Chapters 13, 18, 21).

18.27.3		$$Q_n(x)=Q_n(x;\alpha ,\beta ,N;q)={}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{q^{-n},\alpha \beta q^{n+1},x}{\alpha q,q^{-N}};q,q),$$
$n=0,1,\mathrm{\dots },N$.
ⓘ Defines: $Q_n(x;\alpha ,\beta ,N;q)$: $q$-Hahn polynomial Symbols: ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.27(ii) Permalink: http://dlmf.nist.gov/18.27.E3 Encodings: TeX, pMML, png See also: Annotations for §18.27(ii), §18.27 and Ch.18

18.27.4		$$\sum _{y=0}^N{Q}_n(q^{-y})Q_m(q^{-y}){\left[\genfrac{}{}{0.0pt}{}{N}{y}\right]}_q\frac{{(\alpha q;q)}_y{(\beta q;q)}_{N-y}}{{\left(\alpha q\right)}^y}=h_n{\delta }_{n,m},$$
$n,m=0,1,\mathrm{\dots },N$,
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-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: §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 §18.27(ii), §18.27 and Ch.18

with

18.27.4_1		$$h_n=\frac{{\left(\alpha q\right)}^{nN}}{1-\alpha \beta q^{2n+1}}\frac{{(\alpha \beta q^{n+1};q)}_{N+1}{(\beta q;q)}_n}{{\left[\genfrac{}{}{0.0pt}{}{N}{n}\right]}_q{(\alpha q;q)}_n}.$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $h_n$ Referenced by: §18.27(ii), Erratum (V1.2.0) §18.27 Permalink: http://dlmf.nist.gov/18.27.E4_1 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.27(ii), §18.27 and Ch.18

18.27.4_2		$$\underset{q\to 1}{lim}{Q}_n(q^{-x};q^{\alpha },q^{\beta },N;q)=Q_n(x;\alpha ,\beta ,N).$$
ⓘ Symbols: $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $Q_n(x;\alpha ,\beta ,N;q)$: $q$-Hahn polynomial, $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.27(ii), Erratum (V1.2.0) §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 §18.27(ii), §18.27 and Ch.18

18.27.5		$$P_n(x;a,b,c;q)={}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{q^{-n},ab{q}^{n+1},x}{aq,cq};q,q).$$
ⓘ Defines: $P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial Symbols: ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: 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 §18.27(iii), §18.27 and Ch.18

Alternative definitions and notations are

18.27.6		$$P_n^{(\alpha ,\beta )}(x;c,d;q)=\begin{array}{l}\frac{c^n{q}^{-(\alpha +1)n}{(q^{\alpha +1},-q^{\alpha +1}{c}^{-1}d;q)}_n}{{(q,-q;q)}_n}\\ \times P_n(q^{\alpha +1}{c}^{-1}x;q^{\alpha },q^{\beta },-q^{\alpha }{c}^{-1}d;q),\end{array}$$
ⓘ Defines: $P_n^{(\alpha ,\beta )}(x;c,d;q)$: big $q$-Jacobi polynomial and $P_n^{(\alpha ,\beta )}(x;c,d;q)$ (locally) Symbols: $(a,b)$: open interval, $P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: 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), §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 §18.27(iii), §18.27 and Ch.18

and

18.27.6_5		$$P_n(x;a,b,c,d;q)=P_n(qa{c}^{-1}x;a,b,-a{c}^{-1}d;q).$$
ⓘ Symbols: $P_n(x;a,b,c;q)$: 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. 442) Referenced by: §18.27(iii), Erratum (V1.2.0) §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 §18.27(iii), §18.27 and Ch.18

The orthogonality relations are given by (18.27.2), with

18.27.7		$$p_n(x)=P_n(x;a,b,c;q),$$
ⓘ Symbols: $P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial, $p_n(x)$: polynomial of degree $n$, $q$: real variable, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.27.E7 Encodings: TeX, pMML, png See also: Annotations for §18.27(iii), §18.27 and Ch.18

18.27.8		$$X={\{a{q}^{\mathrm{\ell }+1}\}}_{\mathrm{\ell }=0,1,2,\mathrm{\dots }}\cup {\{c{q}^{\mathrm{\ell }+1}\}}_{\mathrm{\ell }=0,1,2,\mathrm{\dots }},$$
ⓘ Symbols: $\cup$: union, $X$: subset of $ℝ$, $q$: real variable and $\mathrm{\ell }$: nonnegative integer Permalink: http://dlmf.nist.gov/18.27.E8 Encodings: TeX, pMML, png See also: Annotations for §18.27(iii), §18.27 and Ch.18

18.27.9		$$v_x=\frac{{(a^{-1}x,c^{-1}x;q)}_{\mathrm{\infty }}}{{(x,b{c}^{-1}x;q)}_{\mathrm{\infty }}},$$
, , ,
ⓘ Symbols: ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: 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 §18.27(iii), §18.27 and Ch.18

18.27.9_5		$$h_n=\frac{{\left(-c\right)}^n{a}^{n+1}}{1-ab{q}^{2n+1}}\frac{{(q;q)}_n{q}^{\left(\genfrac{}{}{0.0pt}{}{n+2}{2}\right)}}{{(aq,cq;q)}_n}\frac{{(q,c^{-1}aq,a^{-1}c,ab{q}^{n+1};q)}_{\mathrm{\infty }}}{{(aq,cq,b{q}^{n+1},c^{-1}ab{q}^{n+1};q)}_{\mathrm{\infty }}},$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $n$: nonnegative integer and $h_n$ Referenced by: §18.27(iii), Erratum (V1.2.0) §18.27 Permalink: http://dlmf.nist.gov/18.27.E9_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.27(iii), §18.27 and Ch.18

and

18.27.10		$$p_n(x)=P_n^{(\alpha ,\beta )}(x;c,d;q)$$
ⓘ Symbols: $P_n^{(\alpha ,\beta )}(x;c,d;q)$: big $q$-Jacobi polynomial, $p_n(x)$: polynomial of degree $n$, $q$: real variable, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.27.E10 Encodings: TeX, pMML, png See also: Annotations for §18.27(iii), §18.27 and Ch.18

18.27.11		$$X={\{c{q}^{\mathrm{\ell }}\}}_{\mathrm{\ell }=0,1,2,\mathrm{\dots }}\cup {\{-d{q}^{\mathrm{\ell }}\}}_{\mathrm{\ell }=0,1,2,\mathrm{\dots }},$$
ⓘ Symbols: $\cup$: union, $X$: subset of $ℝ$, $q$: real variable and $\mathrm{\ell }$: nonnegative integer Permalink: http://dlmf.nist.gov/18.27.E11 Encodings: TeX, pMML, png See also: Annotations for §18.27(iii), §18.27 and Ch.18

18.27.12		$$v_x=\frac{{(qx/c,-qx/d;q)}_{\mathrm{\infty }}}{{(q^{\alpha +1}x/c,-q^{\beta +1}x/d;q)}_{\mathrm{\infty }}},$$
$\alpha ,\beta >-1$, $c,d>0$.
ⓘ Symbols: ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: 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 §18.27(iii), §18.27 and Ch.18

18.27.13		$$\begin{array}{ll}{p}_n(x)& =p_n(x;a,b;q)={}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},ab{q}^{n+1}}{aq};q,qx)\\ & ={(-b)}^{-n}{q}^{-n(n+1)/2}\frac{{(qb;q)}_n}{{(qa;q)}_n}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{q^{-n},ab{q}^{n+1},qbx}{qb,0};q,q).\end{array}$$
ⓘ Defines: $p_n(x;a,b;q)$: little $q$-Jacobi polynomial Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $p_n(x)$: 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 ${}_3{}^{}\varphi _2^{}$ representation was added. See also: Annotations for §18.27(iv), §18.27 and Ch.18

18.27.14		$$\sum _{y=0}^{\mathrm{\infty }}{p}_n(q^y)p_m(q^y)\frac{{(bq;q)}_y{(aq)}^y}{{(q;q)}_y}=h_n{\delta }_{n,m},$$
,
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $y$: real variable, $p_n(x)$: polynomial of degree $n$, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $h_n$ Permalink: http://dlmf.nist.gov/18.27.E14 Encodings: TeX, pMML, png See also: Annotations for §18.27(iv), §18.27 and Ch.18

with

18.27.14_1		$$h_n=\frac{{\left(aq\right)}^n}{1-ab{q}^{2n+1}}\frac{{(q,bq;q)}_n}{{(aq;q)}_n}\frac{{(ab{q}^{n+1};q)}_{\mathrm{\infty }}}{{(aq;q)}_{\mathrm{\infty }}}.$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $n$: nonnegative integer and $h_n$ Source: Koekoek et al. (2010, (14.12.2)) Referenced by: §18.27(iv), (18.28.27), Erratum (V1.2.0) §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 §18.27(iv), §18.27 and Ch.18

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

18.27.15		$$L_n^{(\alpha )}(x;q)=\frac{{(q^{\alpha +1};q)}_n}{{(q;q)}_n}{}_1{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n}}{q^{\alpha +1}};q,-x{q}^{n+\alpha +1}).$$
ⓘ Defines: $L_n^{(\alpha )}(x;q)$: $q$-Laguerre polynomial Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: 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 §18.27(v), §18.27 and Ch.18

The measure is not uniquely determined:

18.27.16		$${\int }_0^{\mathrm{\infty }}{L}_n^{(\alpha )}(x;q)L_m^{(\alpha )}(x;q)\frac{x^{\alpha }}{{(-x;q)}_{\mathrm{\infty }}}dx=\frac{{(q^{\alpha +1};q)}_n}{{(q;q)}_n{q}^n}{h}_0^{(1)}{\delta }_{n,m},$$
$\alpha >-1$,
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$, $\int$: integral, $L_n^{(\alpha )}(x;q)$: $q$-Laguerre polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $h_n$ Permalink: http://dlmf.nist.gov/18.27.E16 Encodings: TeX, pMML, png See also: Annotations for §18.27(v), §18.27 and Ch.18

and

18.27.17		$$\sum _{y=-\mathrm{\infty }}^{\mathrm{\infty }}{L}_n^{(\alpha )}(c{q}^y;q)L_m^{(\alpha )}(c{q}^y;q)\frac{q^{y(\alpha +1)}}{{(-c{q}^y;q)}_{\mathrm{\infty }}}=\frac{{(q^{\alpha +1};q)}_n}{{(q;q)}_n{q}^n}{h}_0^{(2)}{\delta }_{n,m},$$
$\alpha >-1$, $c>0$,
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $L_n^{(\alpha )}(x;q)$: $q$-Laguerre polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $y$: real variable, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $h_n$ Permalink: http://dlmf.nist.gov/18.27.E17 Encodings: TeX, pMML, png See also: Annotations for §18.27(v), §18.27 and Ch.18

with

18.27.17_1		$$h_0^{(1)}=\frac{{(q^{-\alpha };q)}_{\mathrm{\infty }}}{{(q;q)}_{\mathrm{\infty }}}\mathrm{\Gamma }\left(\alpha +1\right)\mathrm{\Gamma }\left(-\alpha \right),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable and $h_n$ Source: Koekoek et al. (2010, (14.21.2)) Referenced by: §18.27(v), Erratum (V1.2.0) §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 §18.27(v), §18.27 and Ch.18

18.27.17_2		$$h_0^{(2)}=\frac{{(q,-c{q}^{\alpha +1},-c^{-1}{q}^{-\alpha };q)}_{\mathrm{\infty }}}{{(q^{\alpha +1},-c,-c^{-1}q;q)}_{\mathrm{\infty }}}.$$
ⓘ Symbols: ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: 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 §18.27(v), §18.27 and Ch.18

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

18.27.18		$$S_n(x;q)=\sum _{\mathrm{\ell }=0}^n\frac{q^{{\mathrm{\ell }}^2}{(-x)}^{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}{(q;q)}_{n-\mathrm{\ell }}}=\frac{1}{{(q;q)}_n}{}_1{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n}}{0};q,-q^{n+1}x).$$
ⓘ Defines: $S_n(x;q)$: Stieltjes–Wigert polynomial Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.27.E18 Encodings: TeX, pMML, png See also: Annotations for §18.27(vi), §18.27 and Ch.18

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

The measure is not uniquely determined:

18.27.19		$${\int }_0^{\mathrm{\infty }}\frac{S_n(x;q)S_m(x;q)}{{(-x,-q{x}^{-1};q)}_{\mathrm{\infty }}}dx=\frac{\ln \left(q^{-1}\right)}{q^n}\frac{{(q;q)}_{\mathrm{\infty }}}{{(q;q)}_n}{\delta }_{n,m},$$
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $S_n(x;q)$: Stieltjes–Wigert polynomial, $dx$: differential of $x$, $\int$: integral, $\ln z$: principal branch of logarithm function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.27.E19 Encodings: TeX, pMML, png See also: Annotations for §18.27(vi), §18.27 and Ch.18

and

18.27.20		$${\int }_0^{\mathrm{\infty }}{S}_n(q^{\frac{1}{2}}x;q)S_m(q^{\frac{1}{2}}x;q)\exp \left(-\frac{{(\ln x)}^2}{2\ln \left(q^{-1}\right)}\right)dx=\frac{\sqrt{2\pi q^{-1}\ln \left(q^{-1}\right)}}{q^n{(q;q)}_n}{\delta }_{n,m}.$$
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $S_n(x;q)$: Stieltjes–Wigert polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\exp z$: exponential function, $\int$: integral, $\ln z$: principal branch of logarithm function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.27.E20 Encodings: TeX, pMML, png See also: Annotations for §18.27(vi), §18.27 and Ch.18