Asymptotics of some integrals involving modified Bessel and hyper-Bessel functions

We investigate the asymptotic expansion of integrals analogous to Ball’s integral $${\int }_0^{\mathrm{\infty }}{\left(\frac{\mathrm{\Gamma }(1+\nu )\mathrm{\mid }{J}_{\nu }(x)\mathrm{\mid }}{(x\mathrm{/}2{)}^{\nu }}\right)}^{\text{\hspace{0.17em}⁣}n}dx$$ for large $n$ in which the Bessel function $J_{\nu }(x)$ is replaced by the modified Bessel functions $I_{\nu }(x)$ and $K_{\nu }(x)$ together with appropriate exponential factors $e^{\mp x}$, respectively. The above integral with $J_{\nu }(x)$ replaced by a hyper-Bessel function of the type recently discussed in Aktas {\it et al.} [The Ramanujan J., 2019] and taken over a finite interval determined by the first positive zero of the function is also considered for $n\to \mathrm{\infty }$. We give the leading asymptotic behaviour of the hyper-Bessel function for $x\to +\mathrm{\infty }$ in an appendix. Numerical examples are given to illustrate the accuracy of the various expansions obtained.