18 Orthogonal Polynomials Notation 18 Orthogonal Polynomials 18.2 General Orthogonal Polynomials

§18.1 Notation

Permalink:: http://dlmf.nist.gov/18.1

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

§18.1(i) Special Notation

Referenced by:: §18.2(ii), §18.2(ii), §18.2(iv)
Permalink:: http://dlmf.nist.gov/18.1.i

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

	real variables.
	complex variable.
	real variable such that , unless stated otherwise.
$\ell,m$	nonnegative integers.
	nonnegative integer, except in §18.30.
	positive integer.
$\mathop{\delta\/}\nolimits\!\left(x-a\right)$	Dirac delta (§1.17).
$\delta$	arbitrary small positive constant.
$p_{n}(x)$	polynomial in of degree .
$p_{{-1}}(x)$	0.
	weight function $(\geq 0)$ on an open interval .
$w_{x}$	weights at points $x\in X$ of a finite or countably infinite subset of $\Real$ .
OP’s	orthogonal polynomials.

¶ -Differences

Defines:: $\delta_{{x}}$ : central difference and $\nabla_{{x}}$ : backward difference
Keywords:: central differences in imaginary direction, difference operators

Forward differences:

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

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

Backward differences:

$\nabla_{{x}}\left(f(x)\right)=f(x)-f(x-1),$

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

Central differences in imaginary direction:

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

$\delta_{{x}}^{{n+1}}\left(f(x)\right)=\delta_{{x}}\left(\delta_{{x}}^{n}(f(x))% \right).$

¶ -Pochhammer Symbol

Keywords:: -Pochhammer symbol

$\left(z;q\right)_{{0}}=1,$

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

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

¶ Infinite -Product

Keywords:: Charlier polynomials, Gegenbauer polynomials, Hahn polynomials, Krawtchouk polynomials, Meixner–Pollaczek polynomials, Meixner polynomials, OP’s, Racah polynomials, Rogers polynomials, Wilson polynomials, continuous Hahn polynomials, continuous dual Hahn polynomials, dual Hahn polynomials, orthogonal polynomials on the unit circle, -product

$\left(z;q\right)_{{\infty}}=\prod_{{j=0}}^{\infty}(1-zq^{j}),$

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

§18.1(ii) Main Functions

Permalink:: http://dlmf.nist.gov/18.1.ii

The main functions treated in this chapter are:

¶ Classical OP’s

Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, Laguerre polynomials, Legendre polynomials, classical orthogonal polynomials, ultraspherical polynomials

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

¶ Hahn Class OP’s

Keywords:: Hahn class orthogonal polynomials

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

¶ Wilson Class OP’s

Keywords:: Wilson class orthogonal polynomials

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

¶ -Hahn Class OP’s

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

¶ Askey–Wilson Class OP’s

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

¶ Other OP’s

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

¶ Classical OP’s in Two Variables

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

§18.1(iii) Other Notations

Keywords:: Chebyshev polynomials, Jacobi polynomials, ultraspherical polynomials
Permalink:: http://dlmf.nist.gov/18.1.iii

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

18.1.1 $\mathop{C^{{(0)}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{2}{n}\mathop{T_{{n}}% \/}\nolimits\!\left(x\right)=\frac{2(n-1)!}{\left(\tfrac{1}{2}\right)_{{n}}}% \mathop{P^{{(-\frac{1}{2},-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(x\right),$ $n=1,2,3,\dots$ .

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.1.E1
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Nor do we consider the shifted Jacobi polynomials:

18.1.2 $\mathop{G_{{n}}\/}\nolimits\!\left(p,q,x\right)=\frac{n!}{\left(n+p\right)_{{n% }}}\mathop{P^{{(p-q,q-1)}}_{{n}}\/}\nolimits\!\left(2x-1\right),$

Defines:: $\mathop{G_{{n}}\/}\nolimits\!\left(p,q,x\right)$ : shifted Jacobi polynomial
Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, : real variable, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.1.E2
Encodings:: TeX, pMML, png

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

18.1.3

$\mathop{C_{{n}}\/}\nolimits\!\left(x\right)=2\mathop{T_{{n}}\/}\nolimits\!% \left(\tfrac{1}{2}x\right),$

$\mathop{S_{{n}}\/}\nolimits\!\left(x\right)=\mathop{U_{{n}}\/}\nolimits\!\left% (\tfrac{1}{2}x\right).$

Defines:: $\mathop{S_{{n}}\/}\nolimits\!\left(x\right)$ : dilated Chebyshev polynomial and $\mathop{C_{{n}}\/}\nolimits\!\left(x\right)$ : dilated Chebyshev polynomial
Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.1.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

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

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