§18.6 Symmetry, Special Values, and Limits to Monomials

Permalink:: http://dlmf.nist.gov/18.6

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

§18.6(i) Symmetry and Special Values

Notes:: For (18.6.1) see Szegö (1975, (5.1.7)). For Table 18.6.1, Rows 2, 4, 10, see Szegö (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

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

¶ Laguerre

Keywords:: Laguerre polynomials

18.6.1 $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(0\right)=\frac{\left(\alpha+1\right)_{{n}}}{n!}.$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial 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

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

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

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{W_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the fourth kind, $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\mathop{V_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the third kind, $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
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 $f_{n}(0)$.
Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, Legendre polynomials, ultraspherical polynomials
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), §18.3, ¶ ‣ §18.5(iii), §18.5(ii), §18.6(i), §18.6(i), §18.6(ii), §18.7(iii), ¶ ‣ §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.6.T1

§18.6(ii) Limits to Monomials

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).
Keywords:: Jacobi polynomials, Laguerre polynomials, ultraspherical polynomials
Permalink:: http://dlmf.nist.gov/18.6.ii

18.6.2 $\lim _{{\alpha\to\infty}}\frac{\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)}{\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(1\right)}=\left(\frac{1+x}{2}\right)^{n},$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : 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

18.6.3 $\lim _{{\beta\to\infty}}\frac{\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)}{\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(-1\right)}=\left(\frac{1-x}{2}\right)^{n},$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : 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

18.6.4 $\lim _{{\lambda\to\infty}}\frac{\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)}{\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(1\right)}=x^{n},$

Symbols:: $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : 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

18.6.5 $\lim _{{\alpha\to\infty}}\frac{\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(\alpha x\right)}{\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(0\right)}=(1-x)^{n}.$

Symbols:: $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : 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

§18.6 Symmetry, Special Values, and Limits to Monomials

Contents

§18.6(i) Symmetry and Special Values

¶ Laguerre

§18.6(ii) Limits to Monomials