Integral representations of the Laplace transform and moments of the modified Bessel function with respect to the order (Q1276305)

The author derives new integral representations for the Laplace transforms \( \int^\infty_0 \nu^n I_\nu(t)e^{-p\nu}d\nu\) for \(p>0\), and also for the limit case \(p\to+0\), i.e. for the moments \(m_n(t)=\int^\infty_0 \nu^n I_\nu(t) d\nu\). Moreover, he finds \(m_n(t)\sim \frac{1}{\sqrt\pi} 2^{n-1}\Gamma \left(\frac{n+1}{2}\right)t^{n/2} e^t\) for \(t\to \infty\), and also higher asymptotic approximations for the moments.