The integral representation of Legendre function or Laguerre

Question

As to Bessel function, there are many integral representations.
The integral representation of Hermite function is: $$ H_n(x)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-t^2/2}(x+it)^n\,dt. $$ Then do Legendre functions have an integral representation?
And Laguerre function?
Thanks.

Answer 1

The Legendre functions can be written as contour integrals. For example, $$P_{\lambda}(z) = P^{0}_{\lambda}(z) = \frac{1}{2\pi i} \int_{1,z} \frac{(t^2-1)^{\lambda}}{ 2^{\lambda}(t-z)^{\lambda+1}} dt$$ where the contour winds around the points 1 and z in the positive direction and does not wind around $−1$. For real $x$, we have, $$P_{s}(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi} (x+\sqrt{x^2-1}\cos \theta)^{s} d\theta = \frac{1}{\pi}\int_{0}^{1} (x+\sqrt{x^2-1}(2t-1))^{s}\frac{1}{\sqrt{t(1-t)}}dt.. (s\in \mathbb C)$$

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The Laguerre polynomials can also be represented in the following form: $$L_{n}^{(\alpha)}(x^2) = \frac{2(-1)^n}{\pi^{1/2}\Gamma(\alpha +\frac{1}{2})n!}\int_{0}^{\infty}\int_{0}^{\pi} (x^2-r^2 +2ixr\cos \phi)^{n}e^{-r^2}\times r^{2\alpha +1}(\sin \phi)^{2\alpha}d\phi dr$$ where $\alpha>-\frac{1}{2}$.

EDIT: For more information also see here in addition to the links mentioned by @J.M. Hope it helps.