An asymptotic estimate of the integral of the product of two modified Bessel functions and a power function (Q1274121)

This paper is concerned with the estimate as \(\rho\uparrow 1\) of the integral \[ I_a^\rho(\mu,\nu)= \int_0^\infty x^{-\rho} K_{i\mu}(x) K_{i\nu}(ax) dx, \] where \(K_{i\mu}(x)\) is the modified Bessel function, \(a>0\), \(\mu>0\) and \(\nu>0\). The author gives the leading term of the asymptotics of this integral, which is useful in probability theory and has been known for a long time only in the particular case \(a=1\).