§18.10 Integral Representations

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

Ultraspherical

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

Legendre

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

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

Ultraspherical

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

Legendre

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

Laguerre

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

Hermite

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

§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)

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

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

Hermite

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