Let n≥1 be an integer, let
When n=1, this is just Mellin transform of the Bessel K function. When n=2, F(x,y) has an explicit form in product of Gamma functions, given by the Parseval formula for Mellin transform.
For general n, I expect some Stirling formula type estimation for F(x,y). I tried with Laplace method but didn’t get anywhere.