18 Orthogonal Polynomials Notation 18 Orthogonal Polynomials 18.2 General Orthogonal Polynomials

§18.1 Notation

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Permalink:: http://dlmf.nist.gov/18.1
See also:: Annotations for Ch.18

§18.1(i) Special Notation
§18.1(ii) Main Functions
§18.1(iii) Other Notations

§18.1(i) Special Notation

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Referenced by:: item 1, §18.2(i), §18.2(iii), §18.2(ii), §18.2(ii), §18.2(iv)
Permalink:: http://dlmf.nist.gov/18.1.i
See also:: Annotations for §18.1 and Ch.18

(For other notation see Notation for the Special Functions.)

$x, y, t$	real variables.
$z(=x+\mathrm{i}y)$	complex variable.
$q$	real variable such that $0<q<1$ , unless stated otherwise.
$k,l,\ell,$	nonnegative integers.
$m$	nonnegative integer, except in §18.39.
$n$	nonnegative integer, except in §18.30.
$N$	positive integer.
$\delta\left(x-a\right)$	Dirac delta (§1.17).
$\delta$	arbitrary small positive constant.
$p_{n}(x)$	polynomial in $x$ of degree $n$ .
$p_{-1}(x)$	$0$ .
$w(x)$	weight function $(\geq 0)$ on an open interval $(a,b)$ .
$w_{x}$	weights $(>0)$ at points $x\in X$ of a finite or countably infinite subset of $\mathbb{R}$ .
OP’s	orthogonal polynomials.
EOP’s	exceptional orthogonal polynomials.

$x$ -Differences

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Defines:: $\nabla_{\NVar{x}}$ : backward difference and $\delta_{\NVar{x}}$ : central difference
Keywords:: backward, central differences in imaginary direction, central in imaginary direction, difference operators, forward
See also:: Annotations for §18.1(i), §18.1 and Ch.18

Forward differences:

	$\displaystyle\Delta_{x}\left(f(x)\right)$	$\displaystyle=f(x+1)-f(x),$
	$\displaystyle\Delta_{x}^{n+1}\left(f(x)\right)$	$\displaystyle=\Delta_{x}\left(\Delta_{x}^{n}(f(x))\right).$

Backward differences:

	$\displaystyle\nabla_{x}\left(f(x)\right)$	$\displaystyle=f(x)-f(x-1),$
	$\displaystyle\nabla_{x}^{n+1}\left(f(x)\right)$	$\displaystyle=\nabla_{x}\left(\nabla_{x}^{n}(f(x))\right).$

Central differences in imaginary direction:

	$\displaystyle\delta_{x}\left(f(x)\right)$	$\displaystyle=\left(f(x+\tfrac{1}{2}\mathrm{i})-f(x-\tfrac{1}{2}\mathrm{i})% \right)/\mathrm{i},$
	$\displaystyle\delta_{x}^{n+1}\left(f(x)\right)$	$\displaystyle=\delta_{x}\left(\delta_{x}^{n}(f(x))\right).$

$q$ -Pochhammer Symbol

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Keywords:: $q$ -Pochhammer symbol
See also:: Annotations for §18.1(i), §18.1 and Ch.18

	$\displaystyle\left(z;q\right)_{0}$	$\displaystyle=1,$
	$\displaystyle\left(z;q\right)_{n}$	$\displaystyle=(1-z)(1-zq)\cdots(1-zq^{n-1}),$

\left(z_{1},\dots,z_{k};q\right)_{n}=\left(z_{1};q\right)_{n}\cdots\left(z_{k}% ;q\right)_{n}.

Infinite $q$ -Product

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Keywords:: Charlier polynomials, Gegenbauer polynomials, Hahn polynomials, Krawtchouk polynomials, Meixner polynomials, Meixner–Pollaczek polynomials, OP’s, Racah polynomials, Rogers polynomials, Wilson polynomials, continuous Hahn polynomials, continuous dual Hahn polynomials, dual Hahn polynomials, orthogonal polynomials, orthogonal polynomials on the unit circle, $q$ -product
See also:: Annotations for §18.1(i), §18.1 and Ch.18

	$\displaystyle\left(z;q\right)_{\infty}$	$\displaystyle=\prod_{j=0}^{\infty}(1-zq^{j}),$
	$\displaystyle\left(z_{1},\dots,z_{k};q\right)_{\infty}$	$\displaystyle=\left(z_{1};q\right)_{\infty}\cdots\left(z_{k};q\right)_{\infty}.$

§18.1(ii) Main Functions

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Permalink:: http://dlmf.nist.gov/18.1.ii
See also:: Annotations for §18.1 and Ch.18

The main functions treated in this chapter are:

Classical OP’s

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Defines:: $L_{\NVar{n}}\left(\NVar{x}\right)=L^{(0)}_{n}\left(x\right)$ : Laguerre polynomial
Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, Laguerre polynomials, Legendre polynomials, classical orthogonal polynomials, generalized, notation, notations, shifted, ultraspherical polynomials
See also:: Annotations for §18.1(ii), §18.1 and Ch.18

Jacobi: $P^{(\alpha,\beta)}_{n}\left(x\right)$ .
Ultraspherical (or Gegenbauer): $C^{(\lambda)}_{n}\left(x\right)$ .
Chebyshev of first, second, third, and fourth kinds: $T_{n}\left(x\right)$ , $U_{n}\left(x\right)$ , $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ .
Shifted Chebyshev of first and second kinds: $T^{*}_{n}\left(x\right)$ , $U^{*}_{n}\left(x\right)$ .
Legendre: $P_{n}\left(x\right)$ .
Shifted Legendre: $P^{*}_{n}\left(x\right)$ .
Laguerre: $L^{(\alpha)}_{n}\left(x\right)$ and $L_{n}\left(x\right)=L^{(0)}_{n}\left(x\right)$ . ( $L^{(\alpha)}_{n}\left(x\right)$ with $\alpha\neq 0$ is also called Generalized Laguerre.)
Hermite: $H_{n}\left(x\right)$ , $\mathit{He}_{n}\left(x\right)$ .

Hahn Class OP’s

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Keywords:: Hahn class orthogonal polynomials, notation
See also:: Annotations for §18.1(ii), §18.1 and Ch.18

Hahn: $Q_{n}\left(x;\alpha,\beta,N\right)$ .
Krawtchouk: $K_{n}\left(x;p,N\right)$ .
Meixner: $M_{n}\left(x;\beta,c\right)$ .
Charlier: $C_{n}\left(x;a\right)$ .
Continuous Hahn: $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)$ .
Meixner–Pollaczek: $P^{(\lambda)}_{n}\left(x;\phi\right)$ .

Wilson Class OP’s

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Keywords:: Wilson class orthogonal polynomials, notation
See also:: Annotations for §18.1(ii), §18.1 and Ch.18

Wilson: $W_{n}\left(x;a,b,c,d\right)$ .
Racah: $R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)$ .
Continuous Dual Hahn: $S_{n}\left(x;a,b,c\right)$ .
Dual Hahn: $R_{n}\left(x;\gamma,\delta,N\right)$ .

$q$ -Hahn Class OP’s

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See also:: Annotations for §18.1(ii), §18.1 and Ch.18

$q$ -Hahn: $Q_{n}\left(x;\alpha,\beta,N;q\right)$ .
Big $q$ -Jacobi: $P_{n}\left(x;a,b,c;q\right)$ .
Little $q$ -Jacobi: $p_{n}\left(x;a,b;q\right)$ .
$q$ -Laguerre: $L^{(\alpha)}_{n}\left(x;q\right)$ .
Stieltjes–Wigert: $S_{n}\left(x;q\right)$ .
Discrete $q$ -Hermite I: $h_{n}\left(x;q\right)$ .
Discrete $q$ -Hermite II: $\tilde{h}_{n}\left(x;q\right)$ .

Askey–Wilson Class OP’s

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See also:: Annotations for §18.1(ii), §18.1 and Ch.18

Askey–Wilson: $p_{n}\left(x;a,b,c,d\,|\,q\right)$ .
Al-Salam–Chihara: $Q_{n}\left(x;a,b\,|\,q\right)$ .
Continuous $q$ -Ultraspherical: $C_{n}\left(x;\beta\,|\,q\right)$ .
Continuous $q$ -Hermite: $H_{n}\left(x\,|\,q\right)$ .
Continuous $q^{-1}$ -Hermite: $h_{n}\left(x\,|\,q\right)$
$q$ -Racah: $R_{n}\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)$ .

Associated OP’s

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See also:: Annotations for §18.1(ii), §18.1 and Ch.18

Associated OP’s are denoted via addition of the letter $c$ at the end of the listing of parameters in their usual notations.

Other OP’s

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See also:: Annotations for §18.1(ii), §18.1 and Ch.18

Bessel: $y_{n}\left(x;a\right)$ .
Pollaczek: $P^{(\lambda)}_{n}\left(x;a,b\right)$ , $P^{{(\lambda)}}_{n}\left(x;a,b,c\right)$ .

Classical OP’s in Two Variables

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See also:: Annotations for §18.1(ii), §18.1 and Ch.18

Disk: $R^{(\alpha)}_{m,n}\left(z\right)$ .
Triangle: $P^{\alpha,\beta,\gamma}_{m,n}\left(x,y\right)$ .

§18.1(iii) Other Notations

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Keywords:: Chebyshev polynomials, Jacobi polynomials, case $\lambda=0$ , dilated, notation, shifted, ultraspherical polynomials
Referenced by:: Erratum (V1.0.10) for References
Permalink:: http://dlmf.nist.gov/18.1.iii
Correction (effective with 1.0.28):: The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ used in Mason and Handscomb (2003), therefore the expressions for $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ , have been interchanged. Since the current definitions are now consistent with Mason and Handscomb (2003), the sentence and reference added in Version Version 1.0.10 (August 7, 2015) has been removed. For further details see Errata.
Addition (effective with 1.0.10):: A sentence and reference to Mason and Handscomb (2003) were added at the end of this subsection.
See also:: Annotations for §18.1 and Ch.18

In Szegő (1975, §4.7) the ultraspherical polynomials $C^{(\lambda)}_{n}\left(x\right)$ are denoted by $P_{n}^{(\lambda)}(x)$ . The ultraspherical polynomials will not be considered for $\lambda=0$ . They are defined in the literature by $C^{(0)}_{0}\left(x\right)=1$ and

18.1.1		$C^{(0)}_{n}\left(x\right)=\frac{2}{n}T_{n}\left(x\right)=\frac{2(n-1)!}{{\left% (\tfrac{1}{2}\right)_{n}}}P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(x\right),$
$n=1,2,3,\dots$ .
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.1.E1 Encodings: TeX, pMML, png See also: Annotations for §18.1(iii), §18.1 and Ch.18

Nor do we consider the shifted Jacobi polynomials:

18.1.2		$G_{n}\left(p,q,x\right)=\frac{n!}{{\left(n+p\right)_{n}}}P^{(p-q,q-1)}_{n}% \left(2x-1\right),$
ⓘ Defines: $G_{\NVar{n}}\left(\NVar{p},\NVar{q},\NVar{x}\right)$ : shifted Jacobi polynomial Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $q$ : real variable, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.1.E2 Encodings: TeX, pMML, png See also: Annotations for §18.1(iii), §18.1 and Ch.18

or the dilated Chebyshev polynomials of the first and second kinds:

18.1.3	$\displaystyle C_{n}\left(x\right)$	$\displaystyle=2T_{n}\left(\tfrac{1}{2}x\right),$
18.1.3	$\displaystyle S_{n}\left(x\right)$	$\displaystyle=U_{n}\left(\tfrac{1}{2}x\right).$
ⓘ Defines: $S_{\NVar{n}}\left(\NVar{x}\right)$ : dilated Chebyshev polynomial and $C_{\NVar{n}}\left(\NVar{x}\right)$ : dilated Chebyshev polynomial Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.1.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.1(iii), §18.1 and Ch.18

In Koekoek et al. (2010) $\delta_{x}$ denotes the operator $\mathrm{i}\delta_{x}$ .

18 Orthogonal Polynomials 18.2 General Orthogonal Polynomials

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