I am looking for a proof of one of the integrals presented in Harry Bateman's *Tables of Integral Transforms*. The specific integral in question is presented on page 42 in chapter 8.9 as equation (3):


$$ \int\_0^{\infty} \left[x^{\nu+\frac{1}{2}} \mathrm{e}^{-\frac{1}{2} x^2} L^{\nu}\_n\!\left(x^2\right) \right]J\_{\nu}\!\left(xy\right) \sqrt{xy}\,\mathrm{d}x = \left(-1\right)^n \mathrm{e}^{-\frac{1}{2} y^2} y^{\nu+\frac{1}{2}} L^{\nu}\_n\!\left(y^2\right) $$


where $J\_{\nu}$ are the Bessel functions of the first kind and $L^{\nu}\_n$ are the associated Laguerre polynomials. This integral is basically the Hankel transform of the function $f\!\left(x\right) = x^{\nu} \mathrm{e}^{-\frac{1}{2} x^2} L^{\nu}\_n\!\left(x^2\right) $ with an additional factor $\sqrt{y}$ (depending on the [definition](http://en.wikipedia.org/wiki/Hankel_transform#Alternative_definition)).


A similar integral also appears in [this](https://mathoverflow.net/q/352304) question, but the integral in question here is a bit different.


The full reference of the table entry is:   

Harry Bateman, *Tables of Integral Transforms*, Volume 2, p. 42 (ch. 8.9, eq. (3))



Reference for proof of an integral from the "Tables of Integral Transforms" involving a Gaussian and a Laguerre polynomial
The question asks for a reference; the body asks for proof. Here's a proof that's really more of a verification. From the proposed formula make a generating formula:
$$\sum\_{n=0}^\infty z^n \int\_0^\infty x^{\nu+1}\exp{(-x^2/2)}L\_n^{\nu}(x^2) J\_{\nu}(xy)\,dx \overset{?}{=}$$
$$ y^{\nu}\exp{(-y^2/2)} \sum\_{n=0}^\infty (-z)^n L\_n^{\nu}(x^2).$$
Use the generating formula for Laguerre polynomials (Gradshteyn and Rhyzik 8.975.1),
$$\sum\_{n=0}^\infty z^n L\_n^{\nu}(x) = (1-z)^{-\nu-1}\exp[-x \ z/(1-z) ].$$
Use G&R 6.631.4
$$\int\_0^\infty x^{\nu+1}\exp{(-ax^2)}J\_{\nu}(xy)dx = \left( \frac{y}{2a} \right)^{\nu} \frac{1}{2a} \exp\left[-y^2/(2a)\right].$$
The prior two formulas are well known. Algebra completes the verification. For a proof, work backwards.


