Asymptotic expansion of Mellin transform of products of modified Bessel function K

Let n1 be an integer, let

F(x,y)=0un(x+y)(Kxy(u))ndu

for x,y0.

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.

Answer

Attribution
Source : Link , Question Author : Ted Mao , Answer Author : Community

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