18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.4 Graphics 18.6 Symmetry, Special Values, and Limits to Monomials

§18.5 Explicit Representations

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Keywords:: Hermite polynomials, Laguerre polynomials, Legendre polynomials, classical orthogonal polynomials, explicit representations
Referenced by:: Erratum (V1.2.0) §18.5
Permalink:: http://dlmf.nist.gov/18.5
See also:: Annotations for Ch.18

§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

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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 Szegő (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
Addition (effective with 1.2.0):: Equation (18.5.4_5) was added.
See also:: Annotations for §18.5 and Ch.18

Chebyshev

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Keywords:: Chebyshev polynomials, explicit representations, relations to other functions, trigonometric functions
See also:: Annotations for §18.5(i), §18.5 and Ch.18

With $x=\cos\theta=\tfrac{1}{2}(z+z^{-1})$ ,

18.5.1	$\displaystyle T_{n}\left(x\right)$	$\displaystyle=\cos\left(n\theta\right)=\tfrac{1}{2}\left(z^{n}+z^{-n}\right),$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $\cos\NVar{z}$ : cosine function, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.16(iii), §18.18(viii), §18.5(i), §18.5(ii), §18.6(i), §18.9(i), §18.9(ii), Erratum (V1.2.0) for Equations (18.5.1), (18.5.2), (18.5.3), (18.5.4) Permalink: http://dlmf.nist.gov/18.5.E1 Encodings: TeX, pMML, png See also: Annotations for §18.5(i), §18.5(i), §18.5 and Ch.18
18.5.2	$\displaystyle U_{n}\left(x\right)$	$\displaystyle=\frac{\sin\left((n+1)\theta\right)}{\sin\theta}=\dfrac{z^{n+1}-z% ^{-n-1}}{z-z^{-1}},$
ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.18(viii), (18.5.4_5), §18.9(i), Erratum (V1.2.0) for Equations (18.5.1), (18.5.2), (18.5.3), (18.5.4) Permalink: http://dlmf.nist.gov/18.5.E2 Encodings: TeX, pMML, png See also: Annotations for §18.5(i), §18.5(i), §18.5 and Ch.18
18.5.3	$\displaystyle V_{n}\left(x\right)$	$\displaystyle=\frac{\cos\left((n+\tfrac{1}{2})\theta\right)}{\cos\left(\tfrac{% 1}{2}\theta\right)}=\dfrac{z^{n+1}+z^{-n}}{z+1},$
ⓘ Symbols: $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $\cos\NVar{z}$ : cosine function, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: (18.5.3), (18.5.4), Erratum (V1.0.28) for Table 18.3.1, Erratum (V1.2.0) for Equations (18.5.1), (18.5.2), (18.5.3), (18.5.4) Permalink: http://dlmf.nist.gov/18.5.E3 Encodings: TeX, pMML, png 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, the right-hand sides of (18.5.3) and (18.5.4) have been swapped. For further details see Errata. See also: Annotations for §18.5(i), §18.5(i), §18.5 and Ch.18
18.5.4	$\displaystyle W_{n}\left(x\right)$	$\displaystyle=\frac{\sin\left((n+\tfrac{1}{2})\theta\right)}{\sin\left(\tfrac{% 1}{2}\theta\right)}=\dfrac{z^{n+1}-z^{-n}}{z-1}.$
ⓘ Symbols: $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.16(iii), (18.5.3), (18.5.4), §18.5(i), §18.5(ii), §18.6(i), §18.9(ii), Erratum (V1.0.28) for Table 18.3.1, Erratum (V1.2.0) for Equations (18.5.1), (18.5.2), (18.5.3), (18.5.4) Permalink: http://dlmf.nist.gov/18.5.E4 Encodings: TeX, pMML, png 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, the right-hand sides of (18.5.3) and (18.5.4) have been swapped. For further details see Errata. See also: Annotations for §18.5(i), §18.5(i), §18.5 and Ch.18

18.5.4_5		${\mathrm{i}}^{n}U_{n}\left(\tfrac{1}{2\mathrm{i}}\right)=F_{n+1}.$
ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\mathrm{i}$ : imaginary unit and $n$ : nonnegative integer Proof sketch: Use (18.5.2) and (26.11.7). Referenced by: §18.5(i), §18.5(i), Erratum (V1.2.0) §18.5 Permalink: http://dlmf.nist.gov/18.5.E4_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(i), §18.5(i), §18.5 and Ch.18

In (18.5.4_5) see §26.11 for the Fibonacci numbers $F_{n}$ .

§18.5(ii) Rodrigues Formulas

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Notes:: In Table 18.5.1, for Rows 2, 3, 9, 10 see Szegő (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), §18.3
Permalink:: http://dlmf.nist.gov/18.5.ii
See also:: Annotations for §18.5 and Ch.18

18.5.5		$p_{n}(x)=\frac{1}{\kappa_{n}w(x)}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}% \left(w(x)(F(x))^{n}\right).$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p_{n}(x)$ : polynomial of degree $n$ , $w(x)$ : weight function, $n$ : nonnegative integer 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), item 3., §18.39(ii), §18.39(ii), Table 18.5.1, §18.9(iii) Permalink: http://dlmf.nist.gov/18.5.E5 Encodings: TeX, pMML, png See also: Annotations for §18.5(ii), §18.5 and Ch.18

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

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

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

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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, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $p_{n}(x)$ : polynomial of degree $n$ , $w(x)$ : weight function, $n$ : nonnegative integer and $x$ : real variable
Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, Laguerre polynomials, Legendre polynomials, Rodrigues formula, Rodrigues formulas, classical orthogonal polynomials, ultraspherical polynomials
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
Referenced by:: §18.18(iii), §18.39(ii), §18.5(ii), §18.5(ii), §18.9(iii), Erratum (V1.0.28) for Table 18.3.1
Permalink:: http://dlmf.nist.gov/18.5.T1
Addition (effective with 1.2.0):: A column containing the weight function was added to this table.
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 5 and 6 have been modified accordingly. For further details see Errata.
See also:: Annotations for §18.5(ii), §18.5 and Ch.18

Related formula:

18.5.6		$L^{(\alpha)}_{n}\left(\frac{1}{x}\right)=\frac{(-1)^{n}}{n!}x^{n+\alpha+1}{% \mathrm{e}}^{\ifrac{1}{x}}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}\left(x^{-% \alpha-1}{\mathrm{e}}^{-\ifrac{1}{x}}\right).$
ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.5(ii) Permalink: http://dlmf.nist.gov/18.5.E6 Encodings: TeX, pMML, png See also: Annotations for §18.5(ii), §18.5 and Ch.18

See (Erdélyi et al., 1953b, §10.9(37)) for a related formula for ultraspherical polynomials.

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

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Keywords:: Jacobi polynomials, classical orthogonal polynomials, confluent hypergeometric functions, generalized hypergeometric functions, hypergeometric function, relations to other functions, ultraspherical polynomials
Notes:: For (18.5.7) see Szegő (1975, (4.21.2)). For (18.5.8) see Szegő (1975, (4.3.2)). For (18.5.9) see Szegő (1975, (4.7.6)). For (18.5.10) see Szegő (1975, (4.7.31)). For (18.5.11) see Andrews et al. (1999, (6.4.11)). For (18.5.12) see Szegő (1975, (5.3.3)). For (18.5.13) see Szegő (1975, (5.5.4)).
Referenced by:: §18.11(i), Table 18.3.1, §18.3
Permalink:: http://dlmf.nist.gov/18.5.iii
Addition (effective with 1.2.0):: Equations (18.5.11_1)–(18.5.11_4) were added.
See also:: Annotations for §18.5 and Ch.18

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

Jacobi

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See also:: Annotations for §18.5(iii), §18.5 and Ch.18

18.5.7		$P^{(\alpha,\beta)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(n+\alpha+% \beta+1\right)_{\ell}}{\left(\alpha+\ell+1\right)_{n-\ell}}}{\ell!\;(n-\ell)!}% \left(\frac{x-1}{2}\right)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}{{}_{2% }F_{1}}\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}{2}\right),$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.42 Referenced by: §18.17(iv), §18.17(vii), (18.27.12_5), Table 18.3.1, §18.5(iii), §18.5(iii), §18.6(ii), §18.9(iii) Permalink: http://dlmf.nist.gov/18.5.E7 Encodings: TeX, pMML, png See also: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

18.5.8		$P^{(\alpha,\beta)}_{n}\left(x\right)=2^{-n}\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0% pt}{}{n+\alpha}{\ell}\genfrac{(}{)}{0.0pt}{}{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+% 1)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}\left(\frac{x+1}{2}\right)^{n}% {{}_{2}F_{1}}\left({-n,-n-\beta\atop\alpha+1};\frac{x-1}{x+1}\right),$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $\ell$ : 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 See also: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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

Ultraspherical

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See also:: Annotations for §18.5(iii), §18.5 and Ch.18

18.5.9		$C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}{{}_{2}F% _{1}}\left({-n,n+2\lambda\atop\lambda+\tfrac{1}{2}};\frac{1-x}{2}\right),$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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 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 See also: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

18.5.10		$C^{(\lambda)}_{n}\left(x\right)=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}% \frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-2\ell)!}(2x)^{n-2% \ell}=(2x)^{n}\frac{{\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-\tfrac% {1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-\lambda-n};\frac{1}{x^{2}}\right),$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : 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 See also: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

18.5.11		$C^{(\lambda)}_{n}\left(\cos\theta\right)=\sum_{\ell=0}^{n}\frac{{\left(\lambda% \right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\cos\left((n-% 2\ell)\theta\right)={\mathrm{e}}^{\mathrm{i}n\theta}\frac{{\left(\lambda\right% )_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda\atop 1-\lambda-n};{\mathrm{e}}^{-2% \mathrm{i}\theta}\right).$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 22.3.12 Referenced by: §18.18(iv), §18.5(iii) Permalink: http://dlmf.nist.gov/18.5.E11 Encodings: TeX, pMML, png See also: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

Chebyshev

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Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §18.5(iii), §18.5 and Ch.18

18.5.11_1		$T_{n}\left(x\right)=\tfrac{1}{2}n\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}% \frac{(-1)^{\ell}(n-\ell-1)!}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=2^{n-1}x^{n}{{}% _{2}F_{1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-n};\frac{1}% {x^{2}}\right),$
$n\geq 1$ ,
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Source: Erdélyi et al. (1953b, 10.11(22)) Referenced by: §18.5(iii), Erratum (V1.2.0) §18.5 Permalink: http://dlmf.nist.gov/18.5.E11_1 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

18.5.11_2		$T_{n}\left(x\right)={{}_{2}F_{1}}\left({-n,n\atop\frac{1}{2}};\frac{1-x}{2}% \right),$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.5.E11_2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

18.5.11_3		$U_{n}\left(x\right)=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{% \ell}(n-\ell)!}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=\left(2x\right)^{n}{{}_{2}F_{% 1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-n};\frac{1}{x^{2}}% \right),$
ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Source: Erdélyi et al. (1953b, 10.11(23)) Permalink: http://dlmf.nist.gov/18.5.E11_3 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

18.5.11_4		$U_{n}\left(x\right)=\left(n+1\right){{}_{2}F_{1}}\left({-n,n+2\atop\frac{3}{2}% };\frac{1-x}{2}\right).$
ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.5(iii), Erratum (V1.2.0) §18.5 Permalink: http://dlmf.nist.gov/18.5.E11_4 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

Laguerre

ⓘ

Keywords:: Laguerre polynomials, relation to confluent hypergeometric functions
See also:: Annotations for §18.5(iii), §18.5 and Ch.18

18.5.12		$L^{(\alpha)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(\alpha+\ell+1% \right)_{n-\ell}}}{(n-\ell)!\;\ell!}(-x)^{\ell}=\frac{{\left(\alpha+1\right)_{% n}}}{n!}{{}_{1}F_{1}}\left({-n\atop\alpha+1};x\right).$
ⓘ Symbols: ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.54 Referenced by: §18.11(i), §18.17(v), §18.17(vi), §18.17(vii), §18.18(iii), §18.21(ii), Table 18.3.1, (18.34.1), §18.34(i), §18.5(iii), §18.6(ii) Permalink: http://dlmf.nist.gov/18.5.E12 Encodings: TeX, pMML, png See also: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

Hermite

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Keywords:: Hermite polynomials, Jacobi polynomials, Laguerre polynomials, classical orthogonal polynomials, generalized hypergeometric functions, orthogonality properties, parameter constraint, parameter constraints, relations to other functions, ultraspherical polynomials
See also:: Annotations for §18.5(iii), §18.5 and Ch.18

18.5.13		$H_{n}\left(x\right)=n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^% {\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}=(2x)^{n}{{}_{2}F_{0}}\left({-\tfrac{1% }{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{x^{2}}\right).$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.3.10 Referenced by: §18.17(v), §18.17(vii), §18.5(iii) Permalink: http://dlmf.nist.gov/18.5.E13 Encodings: TeX, pMML, png See also: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

For corresponding formulas for Chebyshev, Legendre, and the Hermite $\mathit{He}_{n}$ 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 $P^{(\alpha,\beta)}_{n}\left(x\right)$ 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 $P^{(\alpha,\beta)}_{n}\left(x\right)$ we assume throughout this chapter that $\alpha>-1$ and $\beta>-1$ , unless stated otherwise. Similarly in the cases of the ultraspherical polynomials $C^{(\lambda)}_{n}\left(x\right)$ and the Laguerre polynomials $L^{(\alpha)}_{n}\left(x\right)$ 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
Addition (effective with 1.2.0):: Equation (18.5.17_5) was added.
See also:: Annotations for §18.5 and Ch.18

Chebyshev

ⓘ

See also:: Annotations for §18.5(iv), §18.5 and Ch.18

18.5.14	$\displaystyle T_{0}\left(x\right)$	$\displaystyle=1,$
	$\displaystyle T_{1}\left(x\right)$	$\displaystyle=x,$
	$\displaystyle T_{2}\left(x\right)$	$\displaystyle=2x^{2}-1,$
	$\displaystyle T_{3}\left(x\right)$	$\displaystyle=4x^{3}-3x,$
	$\displaystyle T_{4}\left(x\right)$	$\displaystyle=8x^{4}-8x^{2}+1,$
	$\displaystyle T_{5}\left(x\right)$	$\displaystyle=16x^{5}-20x^{3}+5x,$
	$\displaystyle T_{6}\left(x\right)$	$\displaystyle=32x^{6}-48x^{4}+18x^{2}-1.$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : 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 Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iv), §18.5(iv), §18.5 and Ch.18

18.5.15	$\displaystyle U_{0}\left(x\right)$	$\displaystyle=1,$
	$\displaystyle U_{1}\left(x\right)$	$\displaystyle=2x,$
	$\displaystyle U_{2}\left(x\right)$	$\displaystyle=4x^{2}-1,$
	$\displaystyle U_{3}\left(x\right)$	$\displaystyle=8x^{3}-4x,$
	$\displaystyle U_{4}\left(x\right)$	$\displaystyle=16x^{4}-12x^{2}+1,$
	$\displaystyle U_{5}\left(x\right)$	$\displaystyle=32x^{5}-32x^{3}+6x,$
	$\displaystyle U_{6}\left(x\right)$	$\displaystyle=64x^{6}-80x^{4}+24x^{2}-1.$
ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$ : 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 Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iv), §18.5(iv), §18.5 and Ch.18

Legendre

ⓘ

See also:: Annotations for §18.5(iv), §18.5 and Ch.18

18.5.16	$\displaystyle P_{0}\left(x\right)$	$\displaystyle=1,$
	$\displaystyle P_{1}\left(x\right)$	$\displaystyle=x,$
	$\displaystyle P_{2}\left(x\right)$	$\displaystyle=\tfrac{3}{2}x^{2}-\tfrac{1}{2},$
	$\displaystyle P_{3}\left(x\right)$	$\displaystyle=\tfrac{5}{2}x^{3}-\tfrac{3}{2}x,$
	$\displaystyle P_{4}\left(x\right)$	$\displaystyle=\tfrac{35}{8}x^{4}-\tfrac{15}{4}x^{2}+\tfrac{3}{8},$
	$\displaystyle P_{5}\left(x\right)$	$\displaystyle=\tfrac{63}{8}x^{5}-\tfrac{35}{4}x^{3}+\tfrac{15}{8}x,$
	$\displaystyle P_{6}\left(x\right)$	$\displaystyle=\tfrac{231}{16}x^{6}-\tfrac{315}{16}x^{4}+\tfrac{105}{16}x^{2}-% \tfrac{5}{16}.$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : 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 Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iv), §18.5(iv), §18.5 and Ch.18

Laguerre

ⓘ

See also:: Annotations for §18.5(iv), §18.5 and Ch.18

18.5.17	$\displaystyle L_{0}\left(x\right)$	$\displaystyle=1,$
	$\displaystyle L_{1}\left(x\right)$	$\displaystyle=-x+1,$
	$\displaystyle L_{2}\left(x\right)$	$\displaystyle=\tfrac{1}{2}x^{2}-2x+1,$
	$\displaystyle L_{3}\left(x\right)$	$\displaystyle=-\tfrac{1}{6}x^{3}+\tfrac{3}{2}x^{2}-3x+1,$
	$\displaystyle L_{4}\left(x\right)$	$\displaystyle=\tfrac{1}{24}x^{4}-\tfrac{2}{3}x^{3}+3x^{2}-4x+1,$
	$\displaystyle L_{5}\left(x\right)$	$\displaystyle=-\tfrac{1}{120}x^{5}+\tfrac{5}{24}x^{4}-\tfrac{5}{3}x^{3}+5x^{2}% -5x+1,$
	$\displaystyle L_{6}\left(x\right)$	$\displaystyle=\tfrac{1}{720}x^{6}-\tfrac{1}{20}x^{5}+\tfrac{5}{8}x^{4}-\tfrac{% 10}{3}x^{3}+\tfrac{15}{2}x^{2}-6x+1.$
ⓘ Symbols: $L_{\NVar{n}}\left(\NVar{x}\right)=L^{(0)}_{n}\left(x\right)$ : 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 Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iv), §18.5(iv), §18.5 and Ch.18

18.5.17_5	$\displaystyle L^{(\alpha)}_{0}\left(x\right)$	$\displaystyle=1,$
	$\displaystyle L^{(\alpha)}_{1}\left(x\right)$	$\displaystyle=-x+\alpha+1,$
	$\displaystyle L^{(\alpha)}_{2}\left(x\right)$	$\displaystyle=\tfrac{1}{2}x^{2}-(\alpha+2)x+\tfrac{1}{2}(\alpha+1)(\alpha+2),$
	$\displaystyle L^{(\alpha)}_{3}\left(x\right)$	$\displaystyle=-\tfrac{1}{6}x^{3}+\tfrac{1}{2}(\alpha+3)x^{2}-\tfrac{1}{2}{% \left(\alpha+2\right)_{2}}x+\tfrac{1}{6}{\left(\alpha+1\right)_{3}},$
	$\displaystyle L^{(\alpha)}_{4}\left(x\right)$	$\displaystyle=\tfrac{1}{24}x^{4}-\tfrac{1}{6}(\alpha+4)x^{3}+\tfrac{1}{4}{% \left(\alpha+3\right)_{2}}x^{2}-\tfrac{1}{6}{\left(\alpha+2\right)_{3}}x+% \tfrac{1}{24}{\left(\alpha+1\right)_{4}}.$
ⓘ 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) and $x$ : real variable Referenced by: §18.5(iv), Erratum (V1.2.0) §18.5 Permalink: http://dlmf.nist.gov/18.5.E17_5 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iv), §18.5(iv), §18.5 and Ch.18

Hermite

ⓘ

Keywords:: Chebyshev polynomials, Hermite polynomials, Laguerre polynomials, Legendre polynomials, classical orthogonal polynomials, explicit representations
See also:: Annotations for §18.5(iv), §18.5 and Ch.18

18.5.18	$\displaystyle H_{0}\left(x\right)$	$\displaystyle=1,$
	$\displaystyle H_{1}\left(x\right)$	$\displaystyle=2x,$
	$\displaystyle H_{2}\left(x\right)$	$\displaystyle=4x^{2}-2,$
	$\displaystyle H_{3}\left(x\right)$	$\displaystyle=8x^{3}-12x,$
	$\displaystyle H_{4}\left(x\right)$	$\displaystyle=16x^{4}-48x^{2}+12,$
	$\displaystyle H_{5}\left(x\right)$	$\displaystyle=32x^{5}-160x^{3}+120x,$
	$\displaystyle H_{6}\left(x\right)$	$\displaystyle=64x^{6}-480x^{4}+720x^{2}-120.$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : 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 See also: Annotations for §18.5(iv), §18.5(iv), §18.5 and Ch.18

18.5.19	$\displaystyle\mathit{He}_{0}\left(x\right)$	$\displaystyle=1,$
	$\displaystyle\mathit{He}_{1}\left(x\right)$	$\displaystyle=x,$
	$\displaystyle\mathit{He}_{2}\left(x\right)$	$\displaystyle=x^{2}-1,$
	$\displaystyle\mathit{He}_{3}\left(x\right)$	$\displaystyle=x^{3}-3x,$
	$\displaystyle\mathit{He}_{4}\left(x\right)$	$\displaystyle=x^{4}-6x^{2}+3,$
	$\displaystyle\mathit{He}_{5}\left(x\right)$	$\displaystyle=x^{5}-10x^{3}+15x,$
	$\displaystyle\mathit{He}_{6}\left(x\right)$	$\displaystyle=x^{6}-15x^{4}+45x^{2}-15.$
ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : 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 See also: Annotations for §18.5(iv), §18.5(iv), §18.5 and Ch.18

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).

18.4 Graphics 18.6 Symmetry, Special Values, and Limits to Monomials

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§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

Chebyshev

Laguerre

Hermite

§18.5(iv) Numerical Coefficients

Chebyshev

Legendre

Laguerre

Hermite