Abstract

. 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 requires to find the asymptotic form of coefficients in an asymptotic but divergent expansion. 1 Introduction The number of ways of partitioning an n--set into equivalence classes is the familiar Bell number B n [2], with exponential generating function B(z) j X n0 B n z n n! = e e z \Gamma1 : (1) For reference, the first few Bell numbers are 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147. Expanding the generating function yields an exact infinite sum for Bell numbers, B n = e \Gamma1 X k0 k n k! ; (2) a formula first derived by Dobinski in 1877. Finally, using a combinatorial theory of ...

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