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Let $n\ge 1$ be an integer, let

$$F(x,y)=\int_0^\infty u^{n(x+y)} (K_{x-y}(u))^n du$$

for $x,y\ge 0$.

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.

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