18 Orthogonal Polynomials Other Orthogonal Polynomials 18.26 Wilson Class: Continued 18.28 Askey–Wilson Class

§18.27 -Hahn Class

Keywords:: -Hahn class orthogonal polynomials, -hypergeometric function
Referenced by:: §17.3(v)
Permalink:: http://dlmf.nist.gov/18.27

§18.27(i) Introduction
§18.27(ii) -Hahn Polynomials
§18.27(iii) Big -Jacobi Polynomials
§18.27(iv) Little -Jacobi Polynomials
§18.27(v) -Laguerre Polynomials
§18.27(vi) Stieltjes–Wigert Polynomials
§18.27(vii) Discrete -Hermite I and II Polynomials

§18.27(i) Introduction

Keywords:: -Hahn class orthogonal polynomials, -hypergeometric orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.27.i

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

The -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 -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:: : real variable, : nonnegative integer, $p_{n}(x)$ : polynomial of degree , : real variable, : coefficient, : coefficient, : coefficient and $\lambda_{n}$ : eigenvalue
Referenced by:: §18.28(i)
Permalink:: http://dlmf.nist.gov/18.27.E1
Encodings:: TeX, pMML, png

where , , and are independent of , and where the $\lambda_{n}$ are the eigenvalues. In the -Hahn class OP’s the role of the operator $\ifrac{d}{dx}$ in the Jacobi, Laguerre, and Hermite cases is played by the -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 -Hahn class. These families depend on further parameters, in addition to . The generic (top level) cases are the -Hahn polynomials and the big -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 $\sum_{{x\in X}}p_{n}(x)p_{m}(x)\,|x|\,v_{x}=h_{n}\delta_{{n,m}},$

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, $\in$ : element of, : subset of $\Real$ , : nonnegative integer, : nonnegative integer, $p_{n}(x)$ : polynomial of degree , : real variable, $h_{n}$ and $v_{x}$
Referenced by:: §18.27(i), §18.27(iii)
Permalink:: http://dlmf.nist.gov/18.27.E2
Encodings:: TeX, pMML, png

where is given by $X=\{aq^{y}\}_{{y\in I_{+}}}$ or $X=\{aq^{y}\}_{{y\in I_{+}}}\cup\{-bq^{y}\}_{{y\in I_{-}}}$ . Here 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 -hypergeometric representations, followed by their orthogonality properties. For other formulas, including -difference equations, recurrence relations, duality formulas, special cases, and limit relations, see Koekoek et al. (2010, Chapter 3). See also Gasper and Rahman (2004, pp. 195–199, 228–230) and Ismail (2005, Chapters 13, 18, 21).

§18.27(ii) -Hahn Polynomials

Notes:: For (18.27.3), (18.27.4) see Gasper and Rahman (2004, (7.2.21), (7.2.22)) and Ismail (2005, (18.5.1), (18.5.2)).
Keywords:: -Hahn class orthogonal polynomials, -Hahn polynomials
Permalink:: http://dlmf.nist.gov/18.27.ii

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:: $\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N;q\right)$ : -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 -hypergeometric) function, : real variable, : nonnegative integer, : positive integer and : real variable
Referenced by:: §18.27(ii)
Permalink:: http://dlmf.nist.gov/18.27.E3
Encodings:: TeX, pMML, png

18.27.4 $\sum_{{y=0}}^{N}Q_{n}(q^{{-y}})Q_{m}(q^{{-y}})\frac{(\alpha q,q^{{-N}};q)_{y}(% \alpha\beta q)^{{-y}}}{(q,\beta^{{-1}}q^{{-N}};q)_{y}}=h_{n}\delta_{{n,m}},$ $n,m=0,1,\ldots,N$ .

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, : real variable, : real variable, : nonnegative integer, : nonnegative integer, : positive integer and $h_{n}$
Referenced by:: §18.27(ii)
Permalink:: http://dlmf.nist.gov/18.27.E4
Encodings:: TeX, pMML, png

For $h_{n}$ see Koekoek et al. (2010, Eq. (3.6.2)).

§18.27(iii) Big -Jacobi Polynomials

Notes:: See Gasper and Rahman (2004, (7.3.10), (7.3.12)) and Ismail (2005, (18.4.7), (18.4.14)).
Keywords:: big -Jacobi polynomials, -Hahn class orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.27.iii

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 -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 -hypergeometric) function, : real variable, : nonnegative integer and : 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 -Jacobi polynomial
Symbols:: : open interval, $\mathop{P_{{n}}\/}\nolimits\!\left(x;a,b,c;q\right)$ : big -Jacobi polynomial, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real variable, : nonnegative integer, : 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 $p_{n}(x)=\mathop{P_{{n}}\/}\nolimits\!\left(x;a,b,c;q\right),$

Symbols:: $\mathop{P_{{n}}\/}\nolimits\!\left(x;a,b,c;q\right)$ : big -Jacobi polynomial, : real variable, : nonnegative integer, $p_{n}(x)$ : polynomial of degree and : real variable
Permalink:: http://dlmf.nist.gov/18.27.E7
Encodings:: TeX, pMML, png

18.27.8 ${X=\{aq^{{\ell+1}}\}_{{\ell=0,1,2,\ldots}}\cup\{cq^{{\ell+1}}\}_{{\ell=0,1,2,% \ldots}}},$

Symbols:: $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $\cup$ : union, : subset of $\Real$ , : real variable and $\ell$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.27.E8
Encodings:: TeX, pMML, png

18.27.9 $v_{x}=\frac{(a^{{-1}}x,c^{{-1}}x;q)_{\infty}}{(x,bc^{{-1}}x;q)_{\infty}},$ $0<a<q^{{-1}}$ , $0<b<q^{{-1}}$ ,

Symbols:: : real variable, : real variable and $v_{x}$
Permalink:: http://dlmf.nist.gov/18.27.E9
Encodings:: TeX, pMML, png

and

18.27.10 $p_{n}(x)=\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x;c,d;q\right)$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x;c,d;q\right)$ : big -Jacobi polynomial, : real variable, : nonnegative integer, $p_{n}(x)$ : polynomial of degree and : real variable
Permalink:: http://dlmf.nist.gov/18.27.E10
Encodings:: TeX, pMML, png

18.27.11 $X=\{cq^{\ell}\}_{{\ell=0,1,2,\ldots}}\cup\{-dq^{\ell}\}_{{\ell=0,1,2,\ldots}},$

Symbols:: $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $\cup$ : union, : subset of $\Real$ , : real variable and $\ell$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.27.E11
Encodings:: TeX, pMML, png

18.27.12 $v_{x}=\frac{(qx/c,-qx/d;q)_{\infty}}{(q^{{\alpha+1}}x/c,-q^{{\beta+1}}x/d;q)_{% \infty}},$ $\alpha,\beta>-1$ ,

Symbols:: : real variable, : 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. (3.5.2)).

§18.27(iv) Little -Jacobi Polynomials

Notes:: See Gasper and Rahman (2004, (7.3.1), (7.3.3)).
Keywords:: little -Jacobi polynomials, -Hahn class orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.27.iv

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 -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 -hypergeometric) function, : real variable, : nonnegative integer, $p_{n}(x)$ : polynomial of degree and : real variable
Permalink:: http://dlmf.nist.gov/18.27.E13
Encodings:: TeX, pMML, png

18.27.14 $\sum_{{y=0}}^{\infty}p_{n}(q^{y})p_{m}(q^{y})\frac{\left(bq;q\right)_{{y}}(aq)% ^{y}}{\left(q;q\right)_{{y}}}=h_{n}\delta_{{n,m}},$ $0<a<q^{{-1}},b<q^{{-1}}$ .

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real variable, : real variable, : nonnegative integer, : nonnegative integer, $p_{n}(x)$ : polynomial of degree and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.27.E14
Encodings:: TeX, pMML, png

For $h_{n}$ see Koekoek et al. (2010, Eq. (3.12.2)).

§18.27(v) -Laguerre Polynomials

Notes:: See Ismail (2005, (21.8.2), (21.8.4)) and Moak (1981, Theorem 2).
Keywords:: -Hahn class orthogonal polynomials, -Laguerre polynomials
Permalink:: http://dlmf.nist.gov/18.27.v

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

The measure is not uniquely determined:

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

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, : differential of , $\int$ : integral, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x;q\right)$ : -Laguerre polynomial, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real variable, : nonnegative integer, : nonnegative integer, : real variable and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.27.E16
Encodings:: TeX, pMML, png

where $h_{0}^{{(1)}}$ is given in Koekoek et al. (2010, Eq. (3.21.2), and

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

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x;q\right)$ : -Laguerre polynomial, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real variable, : real variable, : nonnegative integer, : nonnegative integer and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.27.E17
Encodings:: TeX, pMML, png

where $h_{0}^{{(2)}}$ is given in Koekoek et al. (2010, Eq. (3.21.3).

§18.27(vi) Stieltjes–Wigert Polynomials

Notes:: See Ismail (2005, (21.8.3), (21.8.46)).
Keywords:: Stieltjes–Wigert polynomials, -Hahn class orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.27.vi

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:: $\left(a;q\right)_{{n}}$ : -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/18.27.E18
Encodings:: TeX, pMML, png

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

The measure is not uniquely determined:

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

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, $\mathop{S_{{n}}\/}\nolimits\!\left(x;q\right)$ : Stieltjes–Wigert polynomial, : differential of , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real variable, : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.27.E19
Encodings:: TeX, pMML, png

and

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

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, $\mathop{S_{{n}}\/}\nolimits\!\left(x;q\right)$ : Stieltjes–Wigert polynomial, : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real variable, : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.27.E20
Encodings:: TeX, pMML, png

§18.27(vii) Discrete -Hermite I and II Polynomials

Notes:: See Al-Salam and Carlitz (1965).
Keywords:: discrete -Hermite I and II polynomials, -Hahn class orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.27.vii

¶ Discrete -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 -Hermite I polynomial
Symbols:: $\left\lfloor x\right\rfloor$ : floor of , $\left(a;q\right)_{{n}}$ : -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/18.27.E21
Encodings:: TeX, pMML, png

18.27.22 $\sum_{{\ell=0}}^{\infty}\left(\mathop{h_{{n}}\/}\nolimits\!\left(q^{\ell};q% \right)\mathop{h_{{m}}\/}\nolimits\!\left(q^{\ell};q\right)+\mathop{h_{{n}}\/}% \nolimits\!\left(-q^{\ell};q\right)\mathop{h_{{m}}\/}\nolimits\!\left(-q^{\ell% };q\right)\right)\*\left(q^{{\ell+1}},-q^{{\ell+1}};q\right)_{{\infty}}q^{\ell% }=\left(q;q\right)_{{n}}\left({q,-1,-q};q\right)_{{\infty}}q^{{n(n-1)/2}}% \delta_{{n,m}}.$

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, $\mathop{h_{{n}}\/}\nolimits\!\left(x;q\right)$ : discrete -Hermite I polynomial, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real variable, $\ell$ : nonnegative integer, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.27.E22
Encodings:: TeX, pMML, png

¶ Discrete -Hermite II

Keywords:: -Hahn class orthogonal polynomials, -hypergeometric function

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 -Hermite II polynomial
Symbols:: $\left\lfloor x\right\rfloor$ : floor of , $\left(a;q\right)_{{n}}$ : -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/18.27.E23
Encodings:: TeX, pMML, png

18.27.24 $\sum_{{\ell=-\infty}}^{\infty}\left(\mathop{\tilde{h}_{{n}}\/}\nolimits\!\left% (cq^{\ell};q\right)\mathop{\tilde{h}_{{m}}\/}\nolimits\!\left(cq^{\ell};q% \right)+\mathop{\tilde{h}_{{n}}\/}\nolimits\!\left(-cq^{\ell};q\right)\mathop{% \tilde{h}_{{m}}\/}\nolimits\!\left(-cq^{\ell};q\right)\right)\frac{q^{\ell}}{% \left(-c^{2}q^{{2\ell}};q^{2}\right)_{{\infty}}}=2\frac{\left(q^{2},-c^{2}q,-c% ^{{-2}}q;q^{2}\right)_{{\infty}}}{\left(q,-c^{2},-c^{{-2}}q^{2};q^{2}\right)_{% {\infty}}}\frac{\left(q;q\right)_{{n}}}{q^{{n^{2}}}}\delta_{{n,m}},$

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, $\mathop{\tilde{h}_{{n}}\/}\nolimits\!\left(x;q\right)$ : discrete -Hermite II polynomial, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real variable, $\ell$ : nonnegative integer, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.27.E24
Encodings:: TeX, pMML, png

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 18.26 Wilson Class: Continued 18.28 Askey–Wilson Class