18 Orthogonal PolynomialsClassical Orthogonal Polynomials18.9 Recurrence Relations and Derivatives18.11 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

Contents

1. §18.10(i) Dirichlet–Mehler-Type Integral Representations
2. §18.10(ii) Laplace-Type Integral Representations
3. §18.10(iii) Contour Integral Representations
4. §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		$$\begin{array}{ll}\frac{P_n^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_n^{(\alpha ,\alpha )}\left(1\right)}& =\frac{C_n^{(\alpha +\frac{1}{2})}\left(\cos \theta \right)}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}\\ & =\frac{2^{\alpha +\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +\frac{1}{2}\right)}{(\sin \theta )}^{-2\alpha }{\int }_0^{\theta }\frac{\cos \left((n+\alpha +\frac{1}{2})\varphi \right)}{{(\cos \varphi -\cos \theta )}^{-\alpha +\frac{1}{2}}}d\varphi ,\end{array}$$
, $\alpha >-\frac{1}{2}$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: 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		$$P_n\left(\cos \theta \right)=\frac{2^{\frac{1}{2}}}{\pi }{\int }_0^{\theta }\frac{\cos \left((n+\frac{1}{2})\varphi \right)}{{(\cos \varphi -\cos \theta )}^{\frac{1}{2}}}d\varphi ,$$
.
ⓘ Symbols: $P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: 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 $P_n^{(\alpha ,\beta )}$ 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		$$\frac{P_n^{(\alpha ,\beta )}\left(\cos \theta \right)}{P_n^{(\alpha ,\beta )}\left(1\right)}=\frac{2\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha -\beta \right)\mathrm{\Gamma }\left(\beta +\frac{1}{2}\right)}{\int }_0^1{\int }_0^{\pi }\begin{array}{l}{\left({(\cos \frac{1}{2}\theta )}^2-r^2{(\sin \frac{1}{2}\theta )}^2+ir\sin \theta \cos \varphi \right)}^n\\ \times {(1-r^2)}^{\alpha -\beta -1}{r}^{2\beta +1}{(\sin \varphi )}^{2\beta }d\varphi dr,\end{array}$$
$\alpha >\beta >-\frac{1}{2}$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\sin z$: 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		$$\begin{array}{ll}\frac{P_n^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_n^{(\alpha ,\alpha )}\left(1\right)}& =\frac{C_n^{(\alpha +\frac{1}{2})}\left(\cos \theta \right)}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}\\ & =\frac{\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }(\alpha +\frac{1}{2})}{\int }_0^{\pi }{(\cos \theta +i\sin \theta \cos \varphi )}^n{(\sin \varphi )}^{2\alpha }d\varphi ,\end{array}$$
$\alpha >-\frac{1}{2}$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: 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		$$P_n\left(\cos \theta \right)=\frac{1}{\pi }{\int }_0^{\pi }{(\cos \theta +i\sin \theta \cos \varphi )}^{n}d\varphi .$$
ⓘ Symbols: $P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\sin z$: 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		$$L_n^{(\alpha )}\left(x^2\right)=\frac{2{(-1)}^n}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +\frac{1}{2}\right)n!}{\int }_0^{\mathrm{\infty }}{\int }_0^{\pi }\begin{array}{l}{(x^2-r^2+2ixr\cos \varphi )}^n{\mathrm{e}}^{-r^2}\\ \times r^{2\alpha +1}{(\sin \varphi )}^{2\alpha }d\varphi dr,\end{array}$$
$\alpha >-\frac{1}{2}$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: 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		$$H_n\left(x\right)=\frac{2^n}{{\pi }^{\frac{1}{2}}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{(x+it)}^n{\mathrm{e}}^{-t^2}dt.$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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		$$p_n(x)=\frac{g_0(x)}{2\pi \mathrm{i}}{\int }_C{\left(g_1(z,x)\right)}^n{g}_2(z,x){(z-c)}^{-1}dz$$
ⓘ 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, $dx$: 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.

$p_n(x)$	$g_0(x)$	$g_1(z,x)$	$g_2(z,x)$	$c$	Conditions
$P_n^{(\alpha ,\beta )}\left(x\right)$	${(1-x)}^{-\alpha }{(1+x)}^{-\beta }$	${\displaystyle \frac{z^2-1}{2(z-x)}}$	${(1-z)}^{\alpha }{(1+z)}^{\beta }$	$x$	$\pm 1$ outside $C$.
$C_n^{(\lambda )}\left(x\right)$	$1$	$z^{-1}$	${(1-2xz+z^2)}^{-\lambda }$	$0$	${\mathrm{e}}^{\pm \mathrm{i}\theta }$ outside $C$ (where $x=\cos \theta$).
$T_n\left(x\right)$	$1$	$z^{-1}$	${\displaystyle \frac{1-xz}{1-2xz+z^2}}$	$0$
$U_n\left(x\right)$	$1$	$z^{-1}$	${(1-2xz+z^2)}^{-1}$	$0$
$P_n\left(x\right)$	$1$	$z^{-1}$	${(1-2xz+z^2)}^{-\frac{1}{2}}$	$0$
$P_n\left(x\right)$	$1$	${\displaystyle \frac{z^2-1}{2(z-x)}}$	$1$	$x$
$L_n^{(\alpha )}\left(x\right)$	${\mathrm{e}}^x{x}^{-\alpha }$	$z{(z-x)}^{-1}$	$z^{\alpha }{\mathrm{e}}^{-z}$	$x$	$0$ outside $C$.
$H_n\left(x\right)/n!$	$1$	$z^{-1}$	${\mathrm{e}}^{2xz-z^2}$	$0$
${\mathrm{𝐻𝑒}}_n\left(x\right)/n!$	$1$	$z^{-1}$	${\mathrm{e}}^{xz-\frac{1}{2}{z}^2}$	$0$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $H_n\left(x\right)$: Hermite polynomial, ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $C_n^{(\lambda )}\left(x\right)$: 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		$$L_n^{(\alpha )}\left(x\right)=\frac{{\mathrm{e}}^x{x}^{-\frac{1}{2}\alpha }}{n!}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{n+\frac{1}{2}\alpha }{J}_{\alpha }\left(2\sqrt{xt}\right)dt,$$
$\alpha >-1$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: 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		$$\begin{array}{ll}{H}_n\left(x\right)& =\frac{{(-2i)}^n{\mathrm{e}}^{x^2}}{{\pi }^{\frac{1}{2}}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}{t}^n{\mathrm{e}}^{2ixt}dt\\ & =\frac{2^{n+1}}{{\pi }^{\frac{1}{2}}}{\mathrm{e}}^{x^2}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}{t}^n\cos \left(2xt-\frac{1}{2}n\pi \right)dt.\end{array}$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: 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

See also §18.17.

18.9 Recurrence Relations and Derivatives18.11 Relations to Other Functions

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