# A question about eigenvalue equation of Hankel transform

When we think about the Fourier transform in two dimensional polar coordinates, the Hankel transform is the transformation with respect to the polar diameter. Now I have a question, why is the following expression invariant for the Hankel transform? And How to prove this equation?

$$\exp \left(-\frac{1}{2} x\right) x^{m / 2} L_{p}^{m}(x)=\frac{1}{2}(-1)^{p} \int_{0}^{+\infty} \mathrm{d} y \exp \left(-\frac{1}{2} y\right) J_{m}(\sqrt{x y}) y^{m / 2} L_{p}^{m}(y)$$

$$L_{p}^{m}(y)$$ is Laguerre function. m, p is order;

$$J_{m}(\sqrt{x y})$$ is Bessel function with order of m.

L. Yu et al., The Laguerre-Gaussian series representation of two-dimensional fractional Fourier transform. Journal of Physics A: Mathematical and General 31, 9353-9357 (1998).

• The first part is trivial (why is). The Hankel transform is its own inverse. Just apply the Hankel transform and you are done. I agree you should ask on math.se Jun 23 at 12:33

A: An explicit proof, of a more general identity, is in On a Hankel Transform Integral containing an Exponential Function and Two Laguerre Polynomials. See Equation (5), and take $$\sigma=0$$, $$m=n$$.
Another special case ($$m=n$$, $$\nu=2\sigma$$) was discussed at MSE, with a reference to a 1936 paper by Watson, An Integral Equation for the Square of a Laguerre Polynomial