§18.27 $q$-Hahn Class

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 $\{p_n(x)\}$, $n=0,1,\mathrm{\dots },N$, 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: $q$: real variable, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$, $x$: 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)

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, $X$: subset of $ℝ$, $m$: nonnegative integer, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$, $x$: real variable and $h_n$ Referenced by: §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)

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. 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).

18.27.3		$$Q_n(x)=Q_n(x;\alpha ,\beta ,N;q)={}_3\varphi _2(\genfrac{}{}{0pt}{}{q^{-n},\alpha \beta q^{n+1},x}{\alpha q,q^{-N}};q,q),$$
$n=0,1,\mathrm{\dots },N$.
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, $q$: 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 See also: Annotations for 18.27(ii)

18.27.4		$$\sum _{y=0}^N{Q}_n(q^{-y})Q_m(q^{-y})\frac{{(\alpha q,q^{-N};q)}_y{(\alpha \beta q)}^{-y}}{{(q,{\beta }^{-1}{q}^{-N};q)}_y}=h_n{\delta }_{n,m},$$
$n,m=0,1,\mathrm{\dots },N$.
Symbols: ${\delta }_{j,k}$: Kronecker delta, $y$: real variable, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer, $N$: positive integer and $h_n$ Referenced by: §18.27(ii) Permalink: http://dlmf.nist.gov/18.27.E4 Encodings: TeX, pMML, png See also: Annotations for 18.27(ii)

For $h_n$ see Koekoek et al. (2010, Eq. (14.6.2)).

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 Permalink: http://dlmf.nist.gov/18.27.E5 Encodings: TeX, pMML, png See also: Annotations for 18.27(iii)

and

18.27.6		$$P_n^{(\alpha ,\beta )}(x;c,d;q)=\begin{array}{l}\frac{c^n{q}^{-(\alpha +1)n}{(q^{\alpha +1},-q^{\alpha +1}{c}^{-1}d;q)}_n}{{(q,-q;q)}_n}\\ \times P_n(q^{\alpha +1}{c}^{-1}dx;q^{\alpha },q^{\beta },-q^{\alpha }{c}^{-1}d;q).\end{array}$$
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 Permalink: http://dlmf.nist.gov/18.27.E6 Encodings: TeX, pMML, png See also: Annotations for 18.27(iii)

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, $q$: real variable, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable Permalink: http://dlmf.nist.gov/18.27.E7 Encodings: TeX, pMML, png See also: Annotations for 18.27(iii)

18.27.8		$$X={\{a{q}^{\mathrm{\ell }+1}\}}_{\mathrm{\ell }=0,1,2,\mathrm{\dots }}\cup {\{c{q}^{\mathrm{\ell }+1}\}}_{\mathrm{\ell }=0,1,2,\mathrm{\dots }},$$
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)

18.27.9		$$v_x=\frac{{(a^{-1}x,c^{-1}x;q)}_{\mathrm{\infty }}}{{(x,b{c}^{-1}x;q)}_{\mathrm{\infty }}},$$
, , ,
Symbols: $q$: real variable, $x$: real variable and $v_x$ Permalink: http://dlmf.nist.gov/18.27.E9 Encodings: TeX, pMML, png See also: Annotations for 18.27(iii)

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, $q$: real variable, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable Permalink: http://dlmf.nist.gov/18.27.E10 Encodings: TeX, pMML, png See also: Annotations for 18.27(iii)

18.27.11		$$X={\{c{q}^{\mathrm{\ell }}\}}_{\mathrm{\ell }=0,1,2,\mathrm{\dots }}\cup {\{-d{q}^{\mathrm{\ell }}\}}_{\mathrm{\ell }=0,1,2,\mathrm{\dots }},$$
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)

18.27.12		$$v_x=\frac{{(qx/c,-qx/d;q)}_{\mathrm{\infty }}}{{(q^{\alpha +1}x/c,-q^{\beta +1}x/d;q)}_{\mathrm{\infty }}},$$
$\alpha ,\beta >-1$, $c,d>0$.
Symbols: $q$: real variable, $x$: real variable and $v_x$ Permalink: http://dlmf.nist.gov/18.27.E12 Encodings: TeX, pMML, png See also: Annotations for 18.27(iii)

For $h_n$ see Koekoek et al. (2010, Eq. (14.5.2)).

18.27.13		$$p_n(x)=p_n(x;a,b;q)={}_2\varphi _1(\genfrac{}{}{0pt}{}{q^{-n},ab{q}^{n+1}}{aq};q,qx).$$
Defines: $p_n(x;a,b;q)$: little $q$-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, $p_n(x)$: polynomial of degree $n$ and $x$: real variable Permalink: http://dlmf.nist.gov/18.27.E13 Encodings: TeX, pMML, png See also: Annotations for 18.27(iv)

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, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ and $h_n$ Permalink: http://dlmf.nist.gov/18.27.E14 Encodings: TeX, pMML, png See also: Annotations for 18.27(iv)

For $h_n$ see Koekoek et al. (2010, Eq. (14.12.2)).

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

18.27.15		$$L_n^{(\alpha )}(x;q)=\frac{{(q^{\alpha +1};q)}_n}{{(q;q)}_n}{}_1\varphi _1(\genfrac{}{}{0pt}{}{q^{-n}}{q^{\alpha +1}};q,-x{q}^{n+\alpha +1}).$$
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)

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)

where $h_0^{(1)}$ is given in Koekoek et al. (2010, Eq. (14.21.2)), 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)

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).

18.27.18		$$S_n(x;q)=\sum _{\mathrm{\ell }=0}^n\frac{q^{{\mathrm{\ell }}^2}{(-x)}^{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}{(q;q)}_{n-\mathrm{\ell }}}=\frac{1}{{(q;q)}_n}{}_1\varphi _1(\genfrac{}{}{0pt}{}{q^{-n}}{0};q,-q^{n+1}x).$$
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)

(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)

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)