  In this paper some new series and integral representations for the Tur\'anian
of modified Bessel functions of the first kind are given, which give new
asymptotic expansions and tight bounds for the Tur\'an determinant in the
question. It is shown that in the case of natural and real order the Tur\'anian
can be represented in a relatively compact form, which yields a uniform upper
bound for the Tur\'an determinant for modified Bessel functions of the first
kind. Our results complement and improve some of the results from the
literature.
