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We present a detailed analysis (monotonicity, convexity, recurrence relation, closed form expression and tight bounds) of a new special function which we call as the generalized Marcum function of the second kind, which is an analogous survival (or reliability) function to the so-called generalized Marcum Q -function or the generalized Marcum function of the first kind (the survival function of the non-central chi distribution). The main difference between these two generalized Marcum functions is that they involve the modified Bessel functions of the first and second kind. In this paper we are going to see how far the properties of the generalized Marcum function of the first kind may be extended to apply to the generalized Marcum function of the second kind. These two (linearly independent) special functions have similar properties, however in case of the generalized Marcum function of the second kind we need a little bit more technical analysis in order to prove our main results. [ABSTRACT FROM AUTHOR]

In this note our aim is to present some monotonicity properties of the product of modified Bessel functions of the first and second kind. Certain bounds for the product of modified Bessel functions of the first and second kind are also obtained. These bounds improve and extend known bounds for the product of modified Bessel functions of the first and second kind of order zero. A new Turán type inequality is also given for the product of modified Bessel functions, and some open problems are stated, which may be of interest for further research. [ABSTRACT FROM AUTHOR]