Let $K_a$ be the modified Bessel function of second kind, with $a>0$ a real number.
Is there a nice expression for $$\int_0^T t^{2a} K_a(t)^2\;dt,$$ where $T < \infty$?
The expression for $K_a$ is a horrible looking infinite sum, so I was hoping for a reference. I tried various books containing tables of such integrals but wasn't able to find the one I wanted.
$t^a K_a(t)$ is a multiple of the Fourier transform of $\frac{1}{(1+s^2)^{a+1/2}}$, hence by Parseval's identity the integral for $T=+\infty$ just depends on the integral: $$ \int_{0}^{+\infty}\frac{ds}{(1+s^2)^{2a+1}}=\frac{\sqrt{\pi}\,\Gamma\left(2a+\frac{1}{2}\right)}{2\,\Gamma(2a+1)} $$ that can be computed by setting $s=\tan(\theta)$ then recalling the properties of Euler's beta function. For large $a$s we have:
$$ \int_{0}^{+\infty}t^{2a}K_a(t)^2\,dt \approx \left(\frac{2a}{e}\right)^{2a}\sqrt{\frac{\pi^3}{8a}}.$$