18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.5 Explicit Representations 18.7 Interrelations and Limit Relations

§18.6 Symmetry, Special Values, and Limits to Monomials

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Permalink:: http://dlmf.nist.gov/18.6
See also:: Annotations for Ch.18

§18.6(i) Symmetry and Special Values
§18.6(ii) Limits to Monomials

§18.6(i) Symmetry and Special Values

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Notes:: For (18.6.1) see Szegő (1975, (5.1.7)). For Table 18.6.1, Rows 2, 4, 10, see Szegő (1975, (4.1.3), (4.1.1), (4.7.4), (2.3.3), (4.7.3), (5.5.5)); the entries in the fourth and fifth columns of Rows 3 and 4 of Table 18.6.1 follow from (18.5.9) combined (for Row 3) with (18.7.2); the other entries of Row 3 of Table 18.6.1 are the case $\alpha=\beta$ of Row 2; Rows 5–8 follow from (18.5.1)–(18.5.4); Row 9 is the case $\alpha=0$ of Row 3.
Permalink:: http://dlmf.nist.gov/18.6.i
See also:: Annotations for §18.6 and Ch.18

For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1.

Laguerre

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Keywords:: Laguerre polynomials, value at $z=0$
See also:: Annotations for §18.6(i), §18.6 and Ch.18

18.6.1		$L^{(\alpha)}_{n}\left(0\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}.$
ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ) and $n$ : nonnegative integer A&S Ref: 22.4.7 (see column for $f_{n}(0)$ ) Referenced by: §18.17(i), §18.6(i), §18.6(ii) Permalink: http://dlmf.nist.gov/18.6.E1 Encodings: TeX, pMML, png See also: Annotations for §18.6(i), §18.6(i), §18.6 and Ch.18

Table 18.6.1: Classical OP’s: symmetry and special values.

$p_{n}(x)$	$p_{n}(-x)$	$p_{n}(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime}(0)$
$P^{(\alpha,\beta)}_{n}\left(x\right)$	$(-1)^{n}P^{(\beta,\alpha)}_{n}\left(x\right)$	$\ifrac{{\left(\alpha+1\right)_{n}}}{n!}$
$P^{(\alpha,\alpha)}_{n}\left(x\right)$	$(-1)^{n}P^{(\alpha,\alpha)}_{n}\left(x\right)$	$\ifrac{{\left(\alpha+1\right)_{n}}}{n!}$	$\ifrac{(-\tfrac{1}{4})^{n}{\left(n+\alpha+1\right)_{n}}}{n!}$	$\ifrac{(-\tfrac{1}{4})^{n}{\left(n+\alpha+1\right)_{n+1}}}{n!}$
$C^{(\lambda)}_{n}\left(x\right)$	$(-1)^{n}C^{(\lambda)}_{n}\left(x\right)$	$\ifrac{{\left(2\lambda\right)_{n}}}{n!}$	$\ifrac{(-1)^{n}{\left(\lambda\right)_{n}}}{n!}$	$\ifrac{2(-1)^{n}{\left(\lambda\right)_{n+1}}}{n!}$
$T_{n}\left(x\right)$	$(-1)^{n}T_{n}\left(x\right)$	$1$	$(-1)^{n}$	$(-1)^{n}(2n+1)$
$U_{n}\left(x\right)$	$(-1)^{n}U_{n}\left(x\right)$	$n+1$	$(-1)^{n}$	$(-1)^{n}(2n+2)$
$V_{n}\left(x\right)$	$(-1)^{n}W_{n}\left(x\right)$	$1$	$(-1)^{n}$	$(-1)^{n}(2n+2)$
$W_{n}\left(x\right)$	$(-1)^{n}V_{n}\left(x\right)$	$2n+1$	$(-1)^{n}$	$(-1)^{n}(2n+2)$
$P_{n}\left(x\right)$	$(-1)^{n}P_{n}\left(x\right)$	$1$	$\ifrac{(-1)^{n}{\left(\tfrac{1}{2}\right)_{n}}}{n!}$	$\ifrac{2(-1)^{n}{\left(\tfrac{1}{2}\right)_{n+1}}}{n!}$
$H_{n}\left(x\right)$	$(-1)^{n}H_{n}\left(x\right)$		$(-1)^{n}{\left(n+1\right)_{n}}$	$2(-1)^{n}{\left(n+1\right)_{n+1}}$
$\mathit{He}_{n}\left(x\right)$	$(-1)^{n}\mathit{He}_{n}\left(x\right)$		$(-\tfrac{1}{2})^{n}{\left(n+1\right)_{n}}$	$(-\tfrac{1}{2})^{n}{\left(n+1\right)_{n+1}}$

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Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre 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, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable
Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, Legendre polynomials, special values, symmetry, ultraspherical polynomials
A&S Ref:: 22.4.1, 22.4.2, 22.4.4, 22.4.5, 22.4.6, 22.4.8 (See columns for $f_{n}(-x)$ , for $f_{n}(1)$ and for $f_{n}(0)$ )
Referenced by:: §10.60(ii), §18.12, §18.15(i), §18.16(v), §18.17(iv), §18.17(viii), §18.18(iv), §18.18(iv), Table 18.3.1, §18.5(iii), §18.5(ii), §18.6(i), §18.6(i), §18.6(ii), §18.7(iii), §18.9(ii), §18.9(ii), Erratum (V1.0.28) for Table 18.3.1
Permalink:: http://dlmf.nist.gov/18.6.T1
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 $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ , having been interchanged, Rows 6 and 7 have been modified accordingly. For further details see Errata.
See also:: Annotations for §18.6(i), §18.6(i), §18.6 and Ch.18

§18.6(ii) Limits to Monomials

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Keywords:: Jacobi polynomials, Laguerre polynomials, limits to monomials, ultraspherical polynomials
Notes:: (18.6.2) follows from (18.6.3) by the second row of Table 18.6.1. (18.6.3) follows from (18.5.7) together with the second row of Table 18.6.1. (18.6.4) follows from (18.5.10) together with the fourth row of Table 18.6.1. (18.6.5) follows from (18.5.12) together with (18.6.1).
Permalink:: http://dlmf.nist.gov/18.6.ii
See also:: Annotations for §18.6 and Ch.18

18.6.2	$\displaystyle\lim_{\alpha\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}% {P^{(\alpha,\beta)}_{n}\left(1\right)}$	$\displaystyle=\left(\frac{1+x}{2}\right)^{n},$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.6(ii) Permalink: http://dlmf.nist.gov/18.6.E2 Encodings: TeX, pMML, png See also: Annotations for §18.6(ii), §18.6 and Ch.18
18.6.3	$\displaystyle\lim_{\beta\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{% P^{(\alpha,\beta)}_{n}\left(-1\right)}$	$\displaystyle=\left(\frac{1-x}{2}\right)^{n},$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.6(ii) Permalink: http://dlmf.nist.gov/18.6.E3 Encodings: TeX, pMML, png See also: Annotations for §18.6(ii), §18.6 and Ch.18
18.6.4	$\displaystyle\lim_{\lambda\to\infty}\frac{C^{(\lambda)}_{n}\left(x\right)}{C^{% (\lambda)}_{n}\left(1\right)}$	$\displaystyle=x^{n},$
ⓘ Symbols: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.18(iv), §18.6(ii) Permalink: http://dlmf.nist.gov/18.6.E4 Encodings: TeX, pMML, png See also: Annotations for §18.6(ii), §18.6 and Ch.18
18.6.5	$\displaystyle\lim_{\alpha\to\infty}\frac{L^{(\alpha)}_{n}\left(\alpha x\right)% }{L^{(\alpha)}_{n}\left(0\right)}$	$\displaystyle=(1-x)^{n}.$
ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.6(ii) Permalink: http://dlmf.nist.gov/18.6.E5 Encodings: TeX, pMML, png See also: Annotations for §18.6(ii), §18.6 and Ch.18

18.5 Explicit Representations 18.7 Interrelations and Limit Relations

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§18.6 Symmetry, Special Values, and Limits to Monomials

Contents

§18.6(i) Symmetry and Special Values

Laguerre

§18.6(ii) Limits to Monomials