On Finite and Infinite Sums of the Modified Bessel Function of the First Kind

This paper derives novel finite-sum expressions for the modified Bessel function of the first kind using properties of the Marcum Q function. Limits to infinite sums give expressions that are equivalent but frequently more compact when compared with results in the literature. The finite-sum expressions are also used to re-phrase an inequality for the probability that a sum of independent symmetric random vectors lies in a symmetric convex set not as the sum of two modified Bessel functions of the first kind but via a single function $${\phantom {0}}_{1}F_{1}$$ .