§18.10 Integral Representations

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

Ultraspherical

 18.10.1 $\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{2}}\Gamma% \left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma\left(\alpha+\frac{1}{2}\right)}% (\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+\tfrac{1}{2}% )\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}}}\mathrm{d}\phi,$ $0<\theta<\pi$, $\alpha>-\tfrac{1}{2}$.

Legendre

 18.10.2 $P_{n}\left(\cos\theta\right)=\frac{2^{\frac{1}{2}}}{\pi}\int_{0}^{\theta}\frac% {\cos\left((n+\tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{\frac{1}{2}}}% \mathrm{d}\phi,$ $0<\theta<\pi$.

Generalizations of (18.10.1) are given in Gasper (1975, (6),(8)) and Koornwinder (1975a, (5.7),(5.8)).

§18.10(ii) Laplace-Type Integral Representations

Jacobi

 18.10.3 $\frac{P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\beta)}_{n}% \left(1\right)}=\frac{2\Gamma\left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma% \left(\alpha-\beta\right)\Gamma\left(\beta+\tfrac{1}{2}\right)}\*\int_{0}^{1}% \int_{0}^{\pi}\left((\cos\tfrac{1}{2}\theta)^{2}-r^{2}(\sin\tfrac{1}{2}\theta)% ^{2}+ir\sin\theta\cos\phi\right)^{n}(1-r^{2})^{\alpha-\beta-1}r^{2\beta+1}(% \sin\phi)^{2\beta}\mathrm{d}\phi\mathrm{d}r,$ $\alpha>\beta>-\tfrac{1}{2}$.

Ultraspherical

 18.10.4 ${\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}}=\frac{\Gamma\left(\alpha+1\right)}{% \pi^{\frac{1}{2}}\Gamma{(\alpha+\tfrac{1}{2})}}\*{\int_{0}^{\pi}(\cos\theta+i% \sin\theta\cos\phi)^{n}\*(\sin\phi)^{2\alpha}\mathrm{d}\phi},$ $\alpha>-\frac{1}{2}$.

Legendre

 18.10.5 $P_{n}\left(\cos\theta\right)=\frac{1}{\pi}\int_{0}^{\pi}(\cos\theta+i\sin% \theta\cos\phi)^{n}\mathrm{d}\phi.$

Laguerre

 18.10.6 $L^{(\alpha)}_{n}\left(x^{2}\right)=\frac{2(-1)^{n}}{\pi^{\frac{1}{2}}\Gamma% \left(\alpha+\tfrac{1}{2}\right)n!}\*\int_{0}^{\infty}\int_{0}^{\pi}{(x^{2}-r^% {2}+2ixr\cos\phi)^{n}}\*e^{-r^{2}}r^{2\alpha+1}(\sin\phi)^{2\alpha}\mathrm{d}% \phi\mathrm{d}r,$ $\alpha>-\frac{1}{2}$.

Hermite

 18.10.7 $H_{n}\left(x\right)=\frac{2^{n}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}(x+% it)^{n}e^{-t^{2}}\mathrm{d}t.$

§18.10(iii) Contour Integral Representations

Table 18.10.1 gives contour integral representations of the form

 18.10.8 $p_{n}(x)=\frac{g_{0}(x)}{2\pi i}\int_{C}\left(g_{1}(z,x)\right)^{n}g_{2}(z,x)(% z-c)^{-1}\mathrm{d}z$ ⓘ 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 $x$, $\int$: integral, $z$: complex variable, $n$: nonnegative integer, $p_{n}(x)$: polynomial of degree $n$, $C$: closed contour and $x$: real variable Referenced by: Table 18.10.1 Permalink: http://dlmf.nist.gov/18.10.E8 Encodings: TeX, pMML, png See also: Annotations for 18.10(iii), 18.10 and 18

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

§18.10(iv) Other Integral Representations

Laguerre

 18.10.9 $L^{(\alpha)}_{n}\left(x\right)=\frac{e^{x}x^{-\frac{1}{2}\alpha}}{n!}\int_{0}^% {\infty}e^{-t}t^{n+\frac{1}{2}\alpha}J_{\alpha}\left(2\sqrt{xt}\right)\mathrm{% d}t,$ $\alpha>-1$.

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

Hermite

 18.10.10 $H_{n}\left(x\right)=\frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}% ^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\mathrm{d}t=\frac{2^{n+1}}{\pi^{\frac{1}{2}}}e% ^{x^{2}}\int_{0}^{\infty}e^{-t^{2}}t^{n}\cos\left(2xt-\tfrac{1}{2}n\pi\right)% \mathrm{d}t.$