Journal of Integer Sequences, Vol. 14 (2011), Article 11.1.1

The r-Bell Numbers

István Mező
Department of Applied Mathematics and Probability Theory
Faculty of Informatics
University of Debrecen
P. O. Box 12
H-4010 Debrecen


The notion of r-Stirling numbers implies the definition of generalized Bell (or r-Bell) numbers. The r-Bell numbers have appeared in several works, but there is no systematic treatise on this topic. In this paper we fill this gap. We discuss the most important combinatorial, algebraic and analytic properties of these numbers, which generalize similar properties of the Bell numbers. Most of these results seem to be new. It turns out that in a paper of Whitehead, these numbers appeared in a very different context. In addition, we study the so-called r-Bell polynomials.

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(Concerned with sequences A000110 A005493 A005494 A045379 A134980.)

Received November 1 2010; revised version received December 19 2010. Published in Journal of Integer Sequences, December 19 2010.

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