# Bell numbers

## Bell numbers

[′bel ‚nəm·bərz]
(mathematics)
The numbers, Bn , that count the total number of partitions of a set with n elements.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Like the Bell numbers which are the sum of each row of S, we will take sum of each row of the k-Stirling matrix [S.sup.(k)] and call this the k-Bell number and denote it by [B.sup.(k)] = {[B.sup.(k).sub.i] | i [greater than or equal to] 0}.
(11) Obviously [B.sup.(0).sub.i] = [B.sub.i] the (original) Bell numbers, and [B.sup.(1).sub.i] = [B.sup.(0).sub.i+1], so [B.sup.(2).sub.i] = [B.sup.(1).sub.i+1]-[B.sup.(1).sub.i] = [B.sup.(0).sub.i+2]-[B.sup.(0).sub.i+1] (i [greater than or equal to] 0) by Theorem 1.
We call it a matrix of general Bell numbers and denote it by [??] = [[b.sub.i,j]] (i, j [greater than or equal to] 1).
The Peirce matrix [[p.sub.i,j]] (i, j [greater than or equal to] 0) was designed to generate Bell numbers.
Notice that the left border is always comprised of Bell numbers, while the right diagonals (r.diag.) of [P.sup.(2)] and [P.sup.(3)] are of 2 and 3-Bell numbers, respectively.
of [P.sup.(k)] are k-Bell numbers; i.e., [p.sup.(k).sub.i,i] = [B.sup.(k).sub.i-1] while the entries of left border are Bell numbers.
Among their topics are basic tools, Stirling and Bell numbers, normal ordering in the Weyl algebra, a generalization of the Weyl algebra, and the q-deformed generalized Weyl algebra.
which consequently yields the Bell numbers [B.sub.n] when x = 1.
The following beautiful integral representation of the Bell numbers [B.sub.n] was first obtained by Cesaro :
This expression was generalized by Mezo  using a kind of generalization of the classical Bell numbers called r-Bell numbers [B.sub.n,r].
Aigner  established the Hankel transform of the classical Bell numbers. A similar identity was obtained by Mezo  for the Hankel transform of the r-Bell numbers.
Note that when [alpha] = 1, we recover from (74) the Hankel transform of the classical Bell numbers of Aigner , the Hankel transform of the r-Bell numbers of Mezo , and the Hankel transform of the noncentral Bell numbers in .

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