18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.9 Recurrence Relations and Derivatives 18.11 Relations to Other Functions

§18.10 Integral Representations

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Keywords:: classical orthogonal polynomials, integral representations
Permalink:: http://dlmf.nist.gov/18.10
See also:: Annotations for Ch.18

§18.10(i) Dirichlet–Mehler-Type Integral Representations
§18.10(ii) Laplace-Type Integral Representations
§18.10(iii) Contour Integral Representations
§18.10(iv) Other Integral Representations

§18.10(i) Dirichlet–Mehler-Type Integral Representations

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Notes:: For (18.10.1) combine (14.12.1) and (14.3.21). For (18.10.2) see Szegő (1975, (4.8.6)).
Permalink:: http://dlmf.nist.gov/18.10.i
See also:: Annotations for §18.10 and Ch.18

Ultraspherical

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Keywords:: integral representations, ultraspherical polynomials
See also:: Annotations for §18.10(i), §18.10 and Ch.18

18.10.1		$\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{2}}\Gamma% \left(\alpha+1\right)}{{\pi}^{\frac{1}{2}}\Gamma\left(\alpha+\frac{1}{2}\right% )}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+\tfrac{1}{% 2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}}}\,\mathrm{d}\phi,$
$0<\theta<\pi$ , $\alpha>-\tfrac{1}{2}$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial and $n$ : nonnegative integer A&S Ref: 22.10.11 Referenced by: §18.10(i), §18.10(i) Permalink: http://dlmf.nist.gov/18.10.E1 Encodings: TeX, pMML, png See also: Annotations for §18.10(i), §18.10(i), §18.10 and Ch.18

Legendre

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Keywords:: Legendre polynomials, integral representations
See also:: Annotations for §18.10(i), §18.10 and Ch.18

18.10.2		$P_{n}\left(\cos\theta\right)=\frac{2^{\frac{1}{2}}}{\pi}\int_{0}^{\theta}\frac% {\cos\left((n+\tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{\frac{1}{2}}}\,% \mathrm{d}\phi,$
$0<\theta<\pi$ .
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $n$ : nonnegative integer A&S Ref: 22.10.13 Referenced by: §18.10(i) Permalink: http://dlmf.nist.gov/18.10.E2 Encodings: TeX, pMML, png See also: Annotations for §18.10(i), §18.10(i), §18.10 and Ch.18

Generalizations of (18.10.1) for $P^{(\alpha,\beta)}_{n}$ are given in Gasper (1975, (6),(8)) and Koornwinder (1975a, (5.7),(5.8)).

§18.10(ii) Laplace-Type Integral Representations

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Notes:: For (18.10.3) see Askey (1975, (4.20)). For (18.10.4) see Andrews et al. (1999, Theorem 6.7.4). For (18.10.5) see Szegő (1975, (4.8.10)). (18.10.6) can be obtained as a limit case of (18.10.3) in view of (18.7.21). (18.10.7) can be obtained as a limit case of (18.10.4) in view of (18.7.23).
Permalink:: http://dlmf.nist.gov/18.10.ii
See also:: Annotations for §18.10 and Ch.18

Jacobi

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Keywords:: Jacobi polynomials, integral representations
See also:: Annotations for §18.10(ii), §18.10 and Ch.18

18.10.3		$\frac{P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\beta)}_{n}% \left(1\right)}=\frac{2\Gamma\left(\alpha+1\right)}{{\pi}^{\frac{1}{2}}\Gamma% \left(\alpha-\beta\right)\Gamma\left(\beta+\tfrac{1}{2}\right)}\*\int_{0}^{1}% \int_{0}^{\pi}\left((\cos\tfrac{1}{2}\theta)^{2}-r^{2}(\sin\tfrac{1}{2}\theta)% ^{2}+ir\sin\theta\cos\phi\right)^{n}(1-r^{2})^{\alpha-\beta-1}r^{2\beta+1}(% \sin\phi)^{2\beta}\,\mathrm{d}\phi\,\mathrm{d}r,$
$\alpha>\beta>-\tfrac{1}{2}$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $n$ : nonnegative integer Referenced by: §18.10(ii) Permalink: http://dlmf.nist.gov/18.10.E3 Encodings: TeX, pMML, png See also: Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

Ultraspherical

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Keywords:: integral representations, ultraspherical polynomials
See also:: Annotations for §18.10(ii), §18.10 and Ch.18

18.10.4		${\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}}=\frac{\Gamma\left(\alpha+1\right)}{{% \pi}^{\frac{1}{2}}\Gamma{(\alpha+\tfrac{1}{2})}}\{\int_{0}^{\pi}(\cos\theta+i% \sin\theta\cos\phi)^{n}\(\sin\phi)^{2\alpha}\,\mathrm{d}\phi},$
$\alpha>-\frac{1}{2}$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial and $n$ : nonnegative integer A&S Ref: 22.10.10 Referenced by: §18.10(ii) Permalink: http://dlmf.nist.gov/18.10.E4 Encodings: TeX, pMML, png See also: Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

Legendre

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See also:: Annotations for §18.10(ii), §18.10 and Ch.18

18.10.5		$P_{n}\left(\cos\theta\right)=\frac{1}{\pi}\int_{0}^{\pi}(\cos\theta+i\sin% \theta\cos\phi)^{n}\,\mathrm{d}\phi.$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $n$ : nonnegative integer A&S Ref: 22.10.12 Referenced by: §18.10(ii) Permalink: http://dlmf.nist.gov/18.10.E5 Encodings: TeX, pMML, png See also: Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

Laguerre

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Keywords:: Laguerre polynomials, integral representations
See also:: Annotations for §18.10(ii), §18.10 and Ch.18

18.10.6		$L^{(\alpha)}_{n}\left(x^{2}\right)=\frac{2(-1)^{n}}{{\pi}^{\frac{1}{2}}\Gamma% \left(\alpha+\tfrac{1}{2}\right)n!}\\int_{0}^{\infty}\int_{0}^{\pi}{(x^{2}-r^% {2}+2ixr\cos\phi)^{n}}\{\mathrm{e}}^{-r^{2}}r^{2\alpha+1}(\sin\phi)^{2\alpha}% \,\mathrm{d}\phi\,\mathrm{d}r,$
$\alpha>-\frac{1}{2}$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\sin\NVar{z}$ : sine function, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.10(ii) Permalink: http://dlmf.nist.gov/18.10.E6 Encodings: TeX, pMML, png See also: Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

Hermite

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Keywords:: Hermite polynomials, integral representations
See also:: Annotations for §18.10(ii), §18.10 and Ch.18

18.10.7		$H_{n}\left(x\right)=\frac{2^{n}}{{\pi}^{\frac{1}{2}}}\int_{-\infty}^{\infty}(x% +it)^{n}{\mathrm{e}}^{-t^{2}}\,\mathrm{d}t.$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.10(ii) Permalink: http://dlmf.nist.gov/18.10.E7 Encodings: TeX, pMML, png See also: Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

§18.10(iii) Contour Integral Representations

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Notes:: These representations follow from corresponding Rodrigues formulas (§18.5(ii)) or generating functions (§18.12); see Szegő (1975, (4.4.6), (4.82.1), (4.8.16), (4.8.1), (5.4.8)).
Referenced by:: §1.10(xi)
Permalink:: http://dlmf.nist.gov/18.10.iii
See also:: Annotations for §18.10 and Ch.18

Table 18.10.1 gives contour integral representations of the form

18.10.8		$p_{n}(x)=\frac{g_{0}(x)}{2\pi\mathrm{i}}\int_{C}\left(g_{1}(z,x)\right)^{n}g_{% 2}(z,x)(z-c)^{-1}\,\mathrm{d}z$
ⓘ Defines: $g_{0}(x)$ : factors (locally), $g_{1}(z,x)$ : factors (locally), $g_{2}(z,x)$ : factor (locally) and $c$ : center (locally) Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $p_{n}(x)$ : polynomial of degree $n$ , $z$ : complex variable, $n$ : nonnegative integer, $C$ : closed contour and $x$ : real variable Referenced by: Table 18.10.1, (18.17.21_1) Permalink: http://dlmf.nist.gov/18.10.E8 Encodings: TeX, pMML, png See also: Annotations for §18.10(iii), §18.10 and Ch.18

for the Jacobi, Laguerre, and Hermite polynomials. Here $C$ is a simple closed contour encircling $z=c$ once in the positive sense.

Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).

$p_{n}(x)$	$g_{0}(x)$	$g_{1}(z,x)$	$g_{2}(z,x)$	$c$	Conditions
$P^{(\alpha,\beta)}_{n}\left(x\right)$	$(1-x)^{-\alpha}(1+x)^{-\beta}$	$\dfrac{z^{2}-1}{2(z-x)}$	$(1-z)^{\alpha}(1+z)^{\beta}$	$x$	$\pm 1$ outside $C$ .
$C^{(\lambda)}_{n}\left(x\right)$	$1$	$z^{-1}$	$(1-2xz+z^{2})^{-\lambda}$	$0$	${\mathrm{e}}^{\pm\mathrm{i}\theta}$ outside $C$ (where $x=\cos\theta$ ).
$T_{n}\left(x\right)$	$1$	$z^{-1}$	$\dfrac{1-xz}{1-2xz+z^{2}}$	$0$
$U_{n}\left(x\right)$	$1$	$z^{-1}$	$(1-2xz+z^{2})^{-1}$	$0$
$P_{n}\left(x\right)$	$1$	$z^{-1}$	$(1-2xz+z^{2})^{-\frac{1}{2}}$	$0$
$P_{n}\left(x\right)$	$1$	$\dfrac{z^{2}-1}{2(z-x)}$	$1$	$x$
$L^{(\alpha)}_{n}\left(x\right)$	${\mathrm{e}}^{x}x^{-\alpha}$	$z(z-x)^{-1}$	$z^{\alpha}{\mathrm{e}}^{-z}$	$x$	$0$ outside $C$ .
$H_{n}\left(x\right)/n!$	$1$	$z^{-1}$	${\mathrm{e}}^{2xz-z^{2}}$	$0$
$\mathit{He}_{n}\left(x\right)/n!$	$1$	$z^{-1}$	${\mathrm{e}}^{xz-\frac{1}{2}z^{2}}$	$0$

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Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second 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, $\cos\NVar{z}$ : 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, $p_{n}(x)$ : polynomial of degree $n$ , $z$ : complex variable, $n$ : nonnegative integer, $g_{0}(x)$ : factors, $g_{1}(z,x)$ : factors, $g_{2}(z,x)$ : factor, $c$ : center, $C$ : closed contour and $x$ : real variable
Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, Laguerre polynomials, Legendre polynomials, integral representations, ultraspherical polynomials
A&S Ref:: 22.10.1, 22.10.2, 22.10.3, 22.10.4, 22.10.5, 22.10.6, 22.10.7, 22.10.9
Referenced by:: §18.10(iii), (18.17.21_1)
Permalink:: http://dlmf.nist.gov/18.10.T1
See also:: Annotations for §18.10(iii), §18.10 and Ch.18

§18.10(iv) Other Integral Representations

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Notes:: For (18.10.9) see Andrews et al. (1999, (6.2.15)). For (18.10.10) see Andrews et al. (1999, (6.1.4)).
Permalink:: http://dlmf.nist.gov/18.10.iv
See also:: Annotations for §18.10 and Ch.18

Laguerre

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Keywords:: Bessel functions, Laguerre polynomials, integral representation of Laguerre polynomials, integral representations
See also:: Annotations for §18.10(iv), §18.10 and Ch.18

18.10.9		$L^{(\alpha)}_{n}\left(x\right)=\frac{{\mathrm{e}}^{x}x^{-\frac{1}{2}\alpha}}{n% !}\int_{0}^{\infty}{\mathrm{e}}^{-t}t^{n+\frac{1}{2}\alpha}J_{\alpha}\left(2% \sqrt{xt}\right)\,\mathrm{d}t,$
$\alpha>-1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.10.14 Referenced by: §18.10(iv) Permalink: http://dlmf.nist.gov/18.10.E9 Encodings: TeX, pMML, png See also: Annotations for §18.10(iv), §18.10(iv), §18.10 and Ch.18

For the Bessel function $J_{\nu}\left(z\right)$ see §10.2(ii).

Hermite

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Keywords:: Hermite polynomials, classical orthogonal polynomials, integral representations
See also:: Annotations for §18.10(iv), §18.10 and Ch.18

18.10.10		$H_{n}\left(x\right)=\frac{(-2i)^{n}{\mathrm{e}}^{x^{2}}}{{\pi}^{\frac{1}{2}}}% \int_{-\infty}^{\infty}{\mathrm{e}}^{-t^{2}}t^{n}{\mathrm{e}}^{2ixt}\,\mathrm{% d}t=\frac{2^{n+1}}{{\pi}^{\frac{1}{2}}}{\mathrm{e}}^{x^{2}}\int_{0}^{\infty}{% \mathrm{e}}^{-t^{2}}t^{n}\cos\left(2xt-\tfrac{1}{2}n\pi\right)\,\mathrm{d}t.$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.10.15 Referenced by: §18.10(iv) Permalink: http://dlmf.nist.gov/18.10.E10 Encodings: TeX, pMML, png See also: Annotations for §18.10(iv), §18.10(iv), §18.10 and Ch.18

§18.10 Integral Representations

Contents

§18.10(i) Dirichlet–Mehler-Type Integral Representations

Ultraspherical

Legendre

§18.10(ii) Laplace-Type Integral Representations

Jacobi

Ultraspherical

Legendre

Laguerre

Hermite

§18.10(iii) Contour Integral Representations

§18.10(iv) Other Integral Representations

Laguerre

Hermite