Bell numbers, their relatives, and algebraic differential equations

We prove that the ordinary generating function of Bell numbers satisfies no algebraic differential equation over C(x) (in fact, over a larger field). We investigate related numbers counting various set partitions (the Uppuluri–Carpenter numbers, the numbers of partitions with j mod i blocks, the Bessel numbers, the numbers of connected partitions, and the numbers of crossing partitions) and prove for their ogf’s analogous results. Recurrences, functional equations, and continued fraction expansions are derived. key words: Bell number; ordinary generating function; algebraic differential equation; set partition; continued fraction; crossing Address: Martin Klazar Department of Applied Mathematics (KAM) Charles University Malostranské náměst́ı 25 118 00 Praha Czech Republic e-mail: [email protected] fax: 420-2-57531014