Given $n\geq1$ an integer, Is it known that $$ x\to x^nK_n(x) $$ is a decreasing function on $(0,\infty)$? I am looking for a reference or a proof.
2026-03-28 11:33:34.1774697614
Monotonicity of Modified Bessel Functions of the Second type
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$$ \frac{\partial}{\partial x}\left[x^{n}K_{n}\left(x\right)\right]=-\frac{1}{2}x^{n-1}\left(xK_{n-1}\left(x\right)-2nK_{n}\left(x\right)+xK_{n+1}\left(x\right)\right)=-x^{n}K_{n-1}\left(x\right). $$ The claim is thus equivalent to $K_{n}\left(x\right)\geq0$ for all $n\geq0$ and $0< x<\infty$.