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Bell numbers, their relatives, and algebraic differential equations. (English) Zbl 1017.05021
Summary: We prove that the ordinary generating function of Bell numbers satisfies no algebraic differential equation over \(\mathbb{C}(x)\) (in fact, over a larger field). We investigate related numbers counting various set partitions (the Uppuluri-Carpenter numbers, the numbers of partitions with \(j\text{ mod }i\) blocks, the Bessel numbers, the numbers of connected partitions, and the numbers of crossing partitions) and prove analogous results for their ordinary generating functions. Recurrences, functional equations, and continued fraction expansions are derived.

MSC:
05A18 Partitions of sets
05A17 Combinatorial aspects of partitions of integers
11B73 Bell and Stirling numbers
34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
11P81 Elementary theory of partitions
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