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The r-Bell Numbers
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István Mező

Department of Applied Mathematics and Probability Theory

Faculty of Informatics

University of Debrecen

P. O. Box 12

H-4010 Debrecen

Hungary

**Abstract:**

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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