# Orthogonality condition for Airy functions

Hi, all. I'll be brief. Can Airy functions [those who solve the differential equation y''-xy=0] be considered orthogonal over some interval? If so, what is their orthogonality condition? Given that the Airy functions have a representation in terms of Bessel functions, I would be inclined to think that the answer is yes; however, is there a compact form for their orthogonality condition? I've looked everywhere without success. Thanks.

I think you want to look at $$y'' = \lambda x y,$$ whose solutions are Airy functions of $r = \lambda^{1/3}x$, and the eigenfunctions will be orthogonal with respect to $$\int_a^b x y_1(x) y_2(x)\,dx.$$ where $\lambda^{1/3}a$ and $\lambda^{1/3}b$ should be zeroes of Airy functions or their derivatives. Unhelpfully $x = 0$ is not a such a zero.