Non-overlapping Partitions, Continued Fractions, Bessel Functions and a Divergent Series
The counting sequence of a special class of set partitions leads to special numbers called Bessel numbers. The corresponding ordinary generating function has a simple continued fraction expansion related to Bessel functions. We determine here the asymptotic form of Bessel numbers and discuss their relation to Bell numbers. The estimation problem is of some methodological interest as it is necessary to find the asymptotic form of coefficients in an asymptotic but divergent expansion.