4

The following integral equation was presented back in the late 30's by Watson and Szego (Journal of the London Mathematical Society) but I cannot access the journal. Any ideas on a proof ?

$$e^{-x} x^{\alpha } L_n^{\alpha }(x){}^2=\int_0^{\infty } J_{2 \alpha }\left(2 \sqrt{x y}\right) e^{-y} y^{\alpha } L_n^{\alpha }(y){}^2 \, dy$$

where $L_n^{\alpha }(x)$ is the generalized Laguerre polynomial and $J_{2 \alpha }(z)$ is the Bessel function of the first kind.

Joel Storch
  • 358
  • Could you provide us with the reference for this formula? – Gary Mar 12 '21 at 07:53
  • 1
    An Integral Equation for the Square of a Laguerre Polynomial, G. N. Watson Journal of the London Mathematical Society, Volume s1-11, Issue 4, October 1936, Pages 256–261, https://doi.org/10.1112/jlms/s1-11.4.256 – Paul Enta Mar 12 '21 at 21:22
  • this is a special case of a more general formula discussed recently at MathOverflow, see https://mathoverflow.net/a/396005/11260 (take $m=n$, $\nu=2\sigma$ in the formula quoted there). A proof by Szegö is here (behind a paywall, you can email me for a copy). – Carlo Beenakker Jun 25 '21 at 07:21

0 Answers0