§18.5 Explicit Representations

Keywords:: Hermite polynomials, Laguerre polynomials, Legendre polynomials, classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.5

§18.5(i) Trigonometric Functions
§18.5(ii) Rodrigues Formulas
§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions
§18.5(iv) Numerical Coefficients

§18.5(i) Trigonometric Functions

Notes:: To verify (18.5.1)–(18.5.4) substitute them in the fourth through seventh rows, respectively, of Table 18.3.1. Alternatively, combine Szegö (1975, (4.1.7), (4.1.8)) with (18.7.5), (18.7.6).
Referenced by:: §18.11(i)
Permalink:: http://dlmf.nist.gov/18.5.i

¶ Chebyshev

Keywords:: Chebyshev polynomials, trigonometric functions

With $x=\mathop{\cos\/}\nolimits\theta$ ,

18.5.1 $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)=\mathop{\cos\/}\nolimits\!\left(n\theta\right),$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.18(viii), §18.5(i), §18.5(ii), §18.6(i), §18.9(i), §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.5.E1
Encodings:: TeX, pMML, png

18.5.2 $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)=\ifrac{(\mathop{\sin\/}\nolimits(n+1)\theta)}{\mathop{\sin\/}\nolimits\theta},$

Symbols:: $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\mathop{\sin\/}\nolimits z$ : sine function, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.18(viii), §18.9(i)
Permalink:: http://dlmf.nist.gov/18.5.E2
Encodings:: TeX, pMML, png

18.5.3 $\mathop{V_{{n}}\/}\nolimits\!\left(x\right)=\ifrac{(\mathop{\sin\/}\nolimits(n+\tfrac{1}{2})\theta)}{\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\theta\right)},$

Symbols:: $\mathop{V_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the third kind, $\mathop{\sin\/}\nolimits z$ : sine function, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.5.E3
Encodings:: TeX, pMML, png

18.5.4 $\mathop{W_{{n}}\/}\nolimits\!\left(x\right)=\ifrac{(\mathop{\cos\/}\nolimits(n+\tfrac{1}{2})\theta)}{\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\theta\right)}.$

Symbols:: $\mathop{W_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the fourth kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.5(i), §18.5(ii), §18.6(i), §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.5.E4
Encodings:: TeX, pMML, png

§18.5(ii) Rodrigues Formulas

Notes:: In Table 18.5.1, for Rows 2, 3, 9, 10 see Szegö (1975, (4.3.1), (4.7.12), (5.1.5), (5.5.3)); Rows 4–7 follow from Row 2 combined with (18.5.1)–(18.5.4) and Table 18.6.1, second row; Row 8 is the case $\alpha=\beta=0$ of Row 2. For (18.5.6) see Truesdell (1948, §18, (5)).
Referenced by:: §15.9(i), §18.10(iii)
Permalink:: http://dlmf.nist.gov/18.5.ii

18.5.5 $p_{n}(x)=\frac{1}{\kappa _{n}w(x)}\frac{{d}^{n}}{{dx}^{n}}\left(w(x)(F(x))^{n}\right).$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $w(x)$ : weight function, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
A&S Ref:: 22.1.6 (incorrectly stated as property of general OP’s)
Referenced by:: §18.17(v), §18.17(vi), §18.17(vii), Table 18.5.1, §18.9(iii)
Permalink:: http://dlmf.nist.gov/18.5.E5
Encodings:: TeX, pMML, png

In this equation $w(x)$ is as in Table 18.3.1, and $F(x)$ , $\kappa _{n}$ are as in Table 18.5.1.

Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).

$p_{n}(x)$	$F(x)$	$\kappa _{n}$
$\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$	$1-x^{2}$	$(-2)^{n}n!$
$\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$	$1-x^{2}$	$\dfrac{(-2)^{n}\left(\lambda+\frac{1}{2}\right)_{{n}}n!}{\left(2\lambda\right)_{{n}}}$
$\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$	$1-x^{2}$	$(-2)^{n}\left(\frac{1}{2}\right)_{{n}}$
$\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$	$1-x^{2}$	$\dfrac{(-2)^{n}\left(\frac{3}{2}\right)_{{n}}}{n+1}$
$\mathop{V_{{n}}\/}\nolimits\!\left(x\right)$	$1-x^{2}$	$\dfrac{(-2)^{n}\left(\frac{3}{2}\right)_{{n}}}{2n+1}$
$\mathop{W_{{n}}\/}\nolimits\!\left(x\right)$	$1-x^{2}$	$(-2)^{n}\left(\frac{1}{2}\right)_{{n}}$
$\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$	$1-x^{2}$	$(-2)^{n}n!$
$\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$	$x$	$n!$
$\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$	1	$(-1)^{n}$
$\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$	1	$(-1)^{n}$

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, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
A&S Ref:: 22.11.1, 22.11.2, 22.11.3, 22.11.4, 22.11.5, 22.11.6, 22.11.7, 22.11.8
Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, Laguerre polynomials, Legendre polynomials, Rodrigues formulas, classical orthogonal polynomials, ultraspherical polynomials
Referenced by:: §18.18(iii), §18.5(ii), §18.5(ii), §18.9(iii)
Permalink:: http://dlmf.nist.gov/18.5.T1

Related formula:

18.5.6 $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(\frac{1}{x}\right)=\frac{(-1)^{n}}{n!}x^{{n+\alpha+1}}e^{{\ifrac{1}{x}}}\frac{{d}^{n}}{{dx}^{n}}\left(x^{{-\alpha-1}}e^{{-\ifrac{1}{x}}}\right).$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.5(ii)
Permalink:: http://dlmf.nist.gov/18.5.E6
Encodings:: TeX, pMML, png

§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

Notes:: For (18.5.7) see Szegö (1975, (4.21.2)). For (18.5.8) see Szegö (1975, (4.3.2)). For (18.5.9) see Szegö (1975, (4.7.6)). For (18.5.10) see Szegö (1975, (4.7.31)). For (18.5.11) see Andrews et al. (1999, (6.4.11)). For (18.5.12) see Szegö (1975, (5.3.3)). For (18.5.13) see Szegö (1975, (5.5.4)).
Keywords:: Jacobi polynomials, classical orthogonal polynomials, confluent hypergeometric functions, generalized hypergeometric functions, hypergeometric function, ultraspherical polynomials
Referenced by:: §18.11(i), Table 18.3.1
Permalink:: http://dlmf.nist.gov/18.5.iii

For the definitions of $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits$ , $\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits$ , and $\mathop{{{}_{{2}}F_{{0}}}\/}\nolimits$ see §16.2.

¶ Jacobi

18.5.7 $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)=\sum _{{\ell=0}}^{n}\frac{\left(n+\alpha+\beta+1\right)_{{\ell}}\left(\alpha+\ell+1\right)_{{n-\ell}}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell}=\frac{\left(\alpha+1\right)_{{n}}}{n!}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}{2}\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.42
Referenced by:: §18.17(vii), §18.3, ¶ ‣ §18.5(iii), §18.5(iii), §18.6(ii)
Permalink:: http://dlmf.nist.gov/18.5.E7
Encodings:: TeX, pMML, png

18.5.8 $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)=2^{{-n}}\sum _{{\ell=0}}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{{n-\ell}}(x+1)^{\ell}=\frac{\left(\alpha+1\right)_{{n}}}{n!}\left(\frac{x+1}{2}\right)^{n}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({-n,-n-\beta\atop\alpha+1};\frac{x-1}{x+1}\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\binom{m}{n}$ : binomial coefficient, $!$ : $n!$ : factorial, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.3.1
Referenced by:: ¶ ‣ §18.5(iii), §18.5(iii)
Permalink:: http://dlmf.nist.gov/18.5.E8
Encodings:: TeX, pMML, png

and two similar formulas by symmetry; compare the second row in Table 18.6.1.

¶ Ultraspherical

18.5.9 $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{\left(2\lambda\right)_{{n}}}{n!}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({-n,n+2\lambda\atop\lambda+\tfrac{1}{2}};\frac{1-x}{2}\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.46
Referenced by:: §18.5(iii), §18.6(i)
Permalink:: http://dlmf.nist.gov/18.5.E9
Encodings:: TeX, pMML, png

18.5.10 $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)=\sum _{{\ell=0}}^{{\left\lfloor n/2\right\rfloor}}\frac{(-1)^{\ell}\left(\lambda\right)_{{n-\ell}}}{\ell!\;(n-2\ell)!}(2x)^{{n-2\ell}}=(2x)^{n}\frac{\left(\lambda\right)_{{n}}}{n!}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-\lambda-n};\frac{1}{x^{2}}\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $\left\lfloor x\right\rfloor$ : floor of $x$ , $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.3.4
Referenced by:: §18.17(vii), §18.5(iii), §18.6(ii)
Permalink:: http://dlmf.nist.gov/18.5.E10
Encodings:: TeX, pMML, png

18.5.11 $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=\sum _{{\ell=0}}^{n}\frac{\left(\lambda\right)_{{\ell}}\left(\lambda\right)_{{n-\ell}}}{\ell!\;(n-\ell)!}\mathop{\cos\/}\nolimits\!\left((n-2\ell)\theta\right)=e^{{in\theta}}\frac{\left(\lambda\right)_{{n}}}{n!}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({-n,\lambda\atop 1-\lambda-n};e^{{-2i\theta}}\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 22.3.12
Referenced by:: §18.5(iii)
Permalink:: http://dlmf.nist.gov/18.5.E11
Encodings:: TeX, pMML, png

¶ Laguerre

Keywords:: Laguerre polynomials

18.5.12 $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)=\sum _{{\ell=0}}^{n}\frac{\left(\alpha+\ell+1\right)_{{n-\ell}}}{(n-\ell)!\;\ell!}(-x)^{\ell}=\frac{\left(\alpha+1\right)_{{n}}}{n!}\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!\left({-n\atop\alpha+1};x\right).$

¶ Hermite

Keywords:: Hermite polynomials, Jacobi polynomials, Laguerre polynomials, classical orthogonal polynomials, generalized hypergeometric functions, ultraspherical polynomials

18.5.13 $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)=n!\sum _{{\ell=0}}^{{\left\lfloor n/2\right\rfloor}}\frac{(-1)^{\ell}(2x)^{{n-2\ell}}}{\ell!\;(n-2\ell)!}=(2x)^{n}\mathop{{{}_{{2}}F_{{0}}}\/}\nolimits\!\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{x^{2}}\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $!$ : $n!$ : factorial, $\left\lfloor x\right\rfloor$ : floor of $x$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.3.10
Referenced by:: §18.17(vii), §18.5(iii)
Permalink:: http://dlmf.nist.gov/18.5.E13
Encodings:: TeX, pMML, png

For corresponding formulas for Chebyshev, Legendre, and the Hermite $\mathop{\mathit{He}_{{n}}\/}\nolimits$ polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11).

Note. The first of each of equations (18.5.7) and (18.5.8) can be regarded as definitions of $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ when the conditions $\alpha>-1$ and $\beta>-1$ are not satisfied. However, in these circumstances the orthogonality property (18.2.1) disappears. For this reason, and also in the interest of simplicity, in the case of the Jacobi polynomials $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ we assume throughout this chapter that $\alpha>-1$ and $\beta>-1$ , unless stated otherwise. Similarly in the cases of the ultraspherical polynomials $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ and the Laguerre polynomials $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ we assume that $\lambda>-\tfrac{1}{2},\lambda\neq 0$ , and $\alpha>-1$ , unless stated otherwise.

§18.5(iv) Numerical Coefficients

Notes:: Apply the recurrence relations (§18.9(i)), with initial values obtained from the values of $k_{n}$ and $\tilde{k}_{n}/k_{n}$ given in Table 18.3.1 with $n=0,1$ .
Referenced by:: §18.3
Permalink:: http://dlmf.nist.gov/18.5.iv

¶ Chebyshev

18.5.14

$\mathop{T_{{0}}\/}\nolimits\!\left(x\right)=1,$

$\mathop{T_{{1}}\/}\nolimits\!\left(x\right)=x,$

$\mathop{T_{{2}}\/}\nolimits\!\left(x\right)=2x^{2}-1,$

$\mathop{T_{{3}}\/}\nolimits\!\left(x\right)=4x^{3}-3x,$

$\mathop{T_{{4}}\/}\nolimits\!\left(x\right)=8x^{4}-8x^{2}+1,$

$\mathop{T_{{5}}\/}\nolimits\!\left(x\right)=16x^{5}-20x^{3}+5x,$

$\mathop{T_{{6}}\/}\nolimits\!\left(x\right)=32x^{6}-48x^{4}+18x^{2}-1.$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.5.E14
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png

18.5.15

$\mathop{U_{{0}}\/}\nolimits\!\left(x\right)=1,$

$\mathop{U_{{1}}\/}\nolimits\!\left(x\right)=2x,$

$\mathop{U_{{2}}\/}\nolimits\!\left(x\right)=4x^{2}-1,$

$\mathop{U_{{3}}\/}\nolimits\!\left(x\right)=8x^{3}-4x,$

$\mathop{U_{{4}}\/}\nolimits\!\left(x\right)=16x^{4}-12x^{2}+1,$

$\mathop{U_{{5}}\/}\nolimits\!\left(x\right)=32x^{5}-32x^{3}+6x,$

$\mathop{U_{{6}}\/}\nolimits\!\left(x\right)=64x^{6}-80x^{4}+24x^{2}-1.$

Symbols:: $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.5.E15
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png

¶ Legendre

18.5.16

$\mathop{P_{{0}}\/}\nolimits\!\left(x\right)=1,$

$\mathop{P_{{1}}\/}\nolimits\!\left(x\right)=x,$

$\mathop{P_{{2}}\/}\nolimits\!\left(x\right)=\tfrac{3}{2}x^{2}-\tfrac{1}{2},$

$\mathop{P_{{3}}\/}\nolimits\!\left(x\right)=\tfrac{5}{2}x^{3}-\tfrac{3}{2}x,$

$\mathop{P_{{4}}\/}\nolimits\!\left(x\right)=\tfrac{35}{8}x^{4}-\tfrac{15}{4}x^{2}+\tfrac{3}{8},$

$\mathop{P_{{5}}\/}\nolimits\!\left(x\right)=\tfrac{63}{8}x^{5}-\tfrac{35}{4}x^{3}+\tfrac{15}{8}x,$

$\mathop{P_{{6}}\/}\nolimits\!\left(x\right)=\tfrac{231}{16}x^{6}-\tfrac{315}{16}x^{4}+\tfrac{105}{16}x^{2}-\tfrac{5}{16}.$

Symbols:: $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.5.E16
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png

¶ Laguerre

18.5.17

$\mathop{L_{{0}}\/}\nolimits\!\left(x\right)=1,$

$\mathop{L_{{1}}\/}\nolimits\!\left(x\right)=-x+1,$

$\mathop{L_{{2}}\/}\nolimits\!\left(x\right)=\tfrac{1}{2}x^{2}-2x+1,$

$\mathop{L_{{3}}\/}\nolimits\!\left(x\right)=-\tfrac{1}{6}x^{3}+\tfrac{3}{2}x^{2}-3x+1,$

$\mathop{L_{{4}}\/}\nolimits\!\left(x\right)=\tfrac{1}{24}x^{4}-\tfrac{2}{3}x^{3}+3x^{2}-4x+1,$

$\mathop{L_{{5}}\/}\nolimits\!\left(x\right)=-\tfrac{1}{120}x^{5}+\tfrac{5}{24}x^{4}-\tfrac{5}{3}x^{3}+5x^{2}-5x+1,$

$\mathop{L_{{6}}\/}\nolimits\!\left(x\right)=\tfrac{1}{720}x^{6}-\tfrac{1}{20}x^{5}+\tfrac{5}{8}x^{4}-\tfrac{10}{3}x^{3}+\tfrac{15}{2}x^{2}-6x+1.$

Symbols:: $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.5.E17
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png

¶ Hermite

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

18.5.18

$\mathop{H_{{0}}\/}\nolimits\!\left(x\right)=1,$

$\mathop{H_{{1}}\/}\nolimits\!\left(x\right)=2x,$

$\mathop{H_{{2}}\/}\nolimits\!\left(x\right)=4x^{2}-2,$

$\mathop{H_{{3}}\/}\nolimits\!\left(x\right)=8x^{3}-12x,$

$\mathop{H_{{4}}\/}\nolimits\!\left(x\right)=16x^{4}-48x^{2}+12,$

$\mathop{H_{{5}}\/}\nolimits\!\left(x\right)=32x^{5}-160x^{3}+120x,$

$\mathop{H_{{6}}\/}\nolimits\!\left(x\right)=64x^{6}-480x^{4}+720x^{2}-120.$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial and $x$ : real variable
Referenced by:: §13.6(v)
Permalink:: http://dlmf.nist.gov/18.5.E18
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png

18.5.19

$\mathop{\mathit{He}_{{0}}\/}\nolimits\!\left(x\right)=1,$

$\mathop{\mathit{He}_{{1}}\/}\nolimits\!\left(x\right)=x,$

$\mathop{\mathit{He}_{{2}}\/}\nolimits\!\left(x\right)=x^{2}-1,$

$\mathop{\mathit{He}_{{3}}\/}\nolimits\!\left(x\right)=x^{3}-3x,$

$\mathop{\mathit{He}_{{4}}\/}\nolimits\!\left(x\right)=x^{4}-6x^{2}+3,$

$\mathop{\mathit{He}_{{5}}\/}\nolimits\!\left(x\right)=x^{5}-10x^{3}+15x,$

$\mathop{\mathit{He}_{{6}}\/}\nolimits\!\left(x\right)=x^{6}-15x^{4}+45x^{2}-15.$

Symbols:: $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.5.E19
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png

For the corresponding polynomials of degrees 7 through 12 see Abramowitz and Stegun (1964, Tables 22.3, 22.5, 22.9, 22.10, 22.12).

§18.5 Explicit Representations

Contents

§18.5(i) Trigonometric Functions

¶ Chebyshev

§18.5(ii) Rodrigues Formulas

§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

¶ Jacobi

¶ Ultraspherical

¶ Laguerre

¶ Hermite

§18.5(iv) Numerical Coefficients

¶ Chebyshev

¶ Legendre

¶ Laguerre

¶ Hermite