The $r$-Bell numbers

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by  Istvan Mezo

The r-Bell numbers
Istva´n Mezo˝1
Department of Applied Mathematics and Probability Theory, Faculty of
0 Informatics, University of Debrecen, Hungary
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n Abstract
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8 ThenotionofgeneralizedBellnumbershasappearedinseveralworksbutthereisno
systematic treatise on this topic. In this paper we fill this gap. We discuss the most
]
O important combinatorial, algebraic and analytic properties of these numbers which
C generalize the 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
h
very different context. In addition, we introduce the so-called r-Bell polynomials.
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[ Key words: Bell numbers, r-Bell numbers, Stirling numbers, r-Stirling numbers

1991 MSC: 11B73
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9 1 Introduction
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: The B Bell number [11] counts the partitions of a set with n elements. Since
v n
i n , for a fixed n and k, enumerates the number of partitions of a set with
X k
r nn eolements consisting k disjoint, nonempty sets, we get immediately that Bn
a
can be given by the sum
n n
B = . (1)
n
k
( )
k=0
X
The numbers n are called Stirling numbers of the second kind. The n-th
k
Bell polynomianl ios
n n
B (x) = xk.
n
k
( )
k=0
X
Email address: [email protected](Istva´n Mez˝o).
URL:
http://www.inf.unideb.hu/valseg/dolgozok/mezoistvan/mezoistvan.html
(Istva´n Mez˝o).
1 Presentaddress:UniversityofDebrecen,H-4010,Debrecen,P.O.Box12,Hungary
Preprint submitted to Elsevier 8 January 2010
These numbers and polynomials have many interesting properties and appear
in several combinatorial identities. A comprehensive paper is [11].
There is a more general version of Stirling numbers, the so-called r-Stirling
numbers. The number n , for fixed n ≥ k ≥ r, enumerates the partitions
k r
of a set of n elements nintoo k nonempty, disjoint subsets such that the first
r elements are in distinct subsets. A systematic treatment on the r-Stirling
numbersisgivenin[4],andadifferent approachisdescribedin[6,7].According
to (1), it seems to be natural to define the numbers
n n+r
B = . (2)
n,r
(k +r)
k=0 r
X
(It is obvious that B = B , because n = n by the definitions.)
n n,0 k k 0
n o n o
The very first question is on the meaning of the r-Bell numbers. By (2), B
n,r
is the number of the partitions of a set with n+r element such that the first
r elements are in distinct subsets in each partition.
The name of r-Stirling numbers suggests the name for the numbers B : we
n,r
call them as r-Bell numbers, and the name of the polynomials
n n+r
B (x) = xk
n,r
(k +r)
k=0 r
X
will be r-Bell polynomials. Thus B = B (1).
n,r n,r
As far as we know, this paper is the first one fully devoted to the r-Bell num-
bers, although Carlitz [6,7] defined these numbers and proved some identities
for them. His original notation was
B = B(n,r).
n,r
2 Example and tables
The following example illuminates again the meaning of the r-Bell numbers.
By definition,
4 4 4
B = + + .
2,2
(2) (3) (4)
2 2 2
4 counts the partitions of 4 element into 2 subsets such that the first 2
2 2
neleoment are in distinct subsets:
{1,3,4},{2} ; {1},{2,3,4} ; {1,3},{2,4} ; {1,4},{2,3}.
2
4 belongs to the partitions
3 2
n o
{1},{2},{3,4} ; {1,3},{2},{4} ; {1,4},{2},{3} ;
{1},{2,3},{4} ; {1},{2,4},{3}.
Finally, 4 equals to the number of partitions of 4 elements into 4 subsets
4 2
(and trivniaolly, the first two elements are in distinct subsets again):
{1},{2},{3},{4}.
That is,
4 4 4
B = + + = 4+5+1 = 10
2,2
(2) (3) (4)
2 2 2
is really the number of all partitions of the set {1,2,3,4} such that the first
two element are in distinct subsets.
The first r-Bell numbers
n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6
r = 0 1 1 2 5 15 52 203
r = 1 1 2 5 15 52 203 877
r = 2 1 3 10 37 151 674 3263
r = 3 1 4 17 77 372 1915 10481
r = 4 1 5 26 141 799 4736 29371
r = 5 1 6 37 235 1540 10427 73013
r = 6 1 7 50 365 2727 20878 163967
The first r-Bell polynomials
B (x)=1
0,r
B (x)=x+r
1,r
B (x)=x2 +(2r+1)x+r2
2,r
B (x)=x3 +(3r+3)x2 +(3r2 +3r+1)x+r3
3,r
B (x)=x4 +(4r+6)x3 +(6r2 +12r+7)x2 +
4,r
(4r3 +6r2 +4r +1)x+r4
3
3 Generating functions
We start to derive the properties of r-Bell numbers. First of all, the generating
functions are determined.
Theorem 3.1 The exponential generating function for the r-Bell polynomials
is
∞ zn
B (x) = ex(ez−1)+rz.
n,r
n!
n=0
X
Proof. Broder [4] gave the double generating function of r-Stirling numbers
∞ n n+r zn
xk = ex(ez−1)+rz.
(k +r) ! n!
n=0 k=0
X X
Althoughhedidnotusetheinnersuminthepastofhispaper,theyareexactly
2
our polynomials. We note that this identity is remarked in [6, eq. (3.19)]
Corollary 3.2 The r-Bell polynomials can be expressed by the usual Bell
ones.
n n
Bn,r(x) = rk Bn−k(x).
k
!
k=0
X
Proof. In the following steps we use the well known identity for ordinary Bell
polynomials:
n n
Bn(x+y) = Bk(x)Bn−k(y).
k
!
k=0
X
Then the exponential generating function of the r-Bell polynomials can be
transformed as
∞ x x zn
B + = e(x2+x2)(ez−1)+rz
n,r
2 2 n!
n=0 (cid:18) (cid:19)
X
∞ zn ∞ x zn ∞ x zn
= rn B B
n n
n=0 n!!" n=0 (cid:18)2(cid:19) n!! n=0 (cid:18)2(cid:19) n!!#
X X X
∞ zn ∞ x x zn
= rn B +
n
n=0 n!! n=0 (cid:18)2 2(cid:19) n!!
X X
∞ n n zn
= rk Bn−k(x) ,
k! ! n!
n=0 k=0
X X
2
by the Cauchy product. Comparing the coefficients, the Corollary follows.
We remark that the non-polynomial version was proven by Carlitz [6, eq.
(3.18)].
4
In order to determine the ordinary generating function we need some other
notions. The falling factorial of a given real number x is denoted and defined
by
xn = x(x−1)(x−2)···(x−n+1), (n = 1,2,...) (3)
and (x)0 = 1, while the rising factorial (a.k.a. Pochhammer symbol) is
(x) ≡ xn = x(x+1)(x+2)···(x+n−1) (n = 1,2,...) (4)
n
with (x) = 1. It is obvious that (1) = n!. Fitting our notations to the theory
0 n
of hypergeometric functions introduced immediately, we apply the notation
(x) instead of xn. The next transformation formula holds
n
xn = (−1)n(−x) . (5)
n
Thehypergeometricfunction(orhypergeometric series) isdefinedbytheseries
F a1, a2, ..., ap t = ∞ (a1)k(a2)k···(ap)k tk.
p q b1, b2, ..., bq (cid:12)(cid:12)(cid:12)  kX=0 (b1)k(b2)k···(bq)k k!
 (cid:12) 
 (cid:12) 
(cid:12)
The ordinary generating function ca(cid:12)n be given with this function.
Theorem 3.3 The r-Bell polynomials have the generating function
∞ rz−1
−1 1
B (x)zn = F z x .
n,r rz −1 ex 1 1 rz+z−1 (cid:12)(cid:12) 
nX=0 z (cid:12)
 (cid:12) 
 (cid:12) 
(cid:12)
(cid:12)
Proof. It is known [4] that for the Stirling numbers
∞ n zm
zn = (m ≥ r ≥ 0).
(m) (1−rz)(1−(r +1)z)···(1−mz)
n=0 r
X
This can be rewritten as
∞ n+r zm
zn = .
n=m(m+r)r (1−rz)(1−(r +1)z)···(1−(m+r)z)
X
We transform the denominator using the falling factorial:
(1−rz)(1−(r +1)z)···(1−(m+r)z)
m+r+1
(1−z)(1−2z)···(1−(m+r)z) zm+1 1
z
= = .
(1−z)(1−2z)···(1−(r −1)z) (cid:16)1(cid:17)r
z
Hence (cid:16) (cid:17)
∞ k +r 1 1 r 1
zk = .
(m+r) z z 1 m+r+1
k=m r (cid:18) (cid:19)
X z
(cid:16) (cid:17)
5
Rule (5) and the definitions (3)-(4) give that
1 m+r+1 1
= (−1)m+r+1 −
z z
(cid:18) (cid:19) (cid:18) (cid:19)m+r+1
1 1
= (−1)m+r+1 − − +r+1 .
z z
(cid:18) (cid:19)r+1(cid:18) (cid:19)m
Consequently,
r
∞ n+r 1 1 (−1)m+r+1
zn = z .
nX=m(m+r)r z −(cid:16)z1(cid:17)r+1 rz+zz−1 m
(cid:16) (cid:17) (cid:16) (cid:17)
Since
r
1
z
z = (−1)r ,
−(cid:16)1(cid:17) rz −1
z r+1
(cid:16) (cid:17)
we get that
∞ n+r −1 (−1)m
zn = .
nX=m(m+r)r rz −1 rz+zz−1 m
Let us multiplication with xm and take a su(cid:16)mmatio(cid:17)n for m over the non-
negative integers,
∞ −1 ∞ (−x)m −1 1
B (x)zn = = F −x .
n,r rz −1 rz+z−1 rz −1 1 1 rz+z−1 (cid:12)(cid:12) 
nX=0 mX=0 z m z (cid:12)
 (cid:12) 
(cid:16) (cid:17)  (cid:12) 
(cid:12)
Finally, we apply Kummer’s formula [1, p. 505.] (cid:12)
a b−a
e−x F x = F −x
1 1 (cid:12)  1 1 (cid:12) 
b (cid:12) b (cid:12)
(cid:12) (cid:12)
 (cid:12)   (cid:12) 
 (cid:12)   (cid:12) 
(cid:12) (cid:12)
with b = rz+z−1 and a = rz−1. (cid:12) 2 (cid:12)
z z
4 Basic recurrences
In an earlier paper of the author [18], the polynomials B (x) were introduced
n,r
because of a very different reason. These functions were used to study the
properties of the r-Stirling numbers and some necessary properties of them
were proven in that paper. We repeat those results without proof.
Theorem 4.1 We have the next recursive identities:
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d
Bn,r(x)=x dxBn−1,r(x)+Bn−1,r(x) +rBn−1,r(x),
!
d
exxrBn,r(x)=xdx (exxrBn−1,r(x)).
Moreover, all roots of B (x) are real and negative.
n,r
A straightforward corollary that for a fixed r the constant term of the n-th
polynomial is rn:
B (0) = rn
n,r
and that the derivative of an r-Bell polynomial is determined by the relation
∂ B (x) rB (x)
B (x) = n+1,r − n,r −B (x).
n,r n,r
∂x x x
The identity
n+r n+r n−1+r
= −(r −1)
(k +r)r (k +r)r−1 ( k +r )r−1
was proven in [4] and implies the recursive relation
Bn,r(x) = Bn,r−1(x)−(r −1)Bn−1,r−1(x).
Finally, we cite Carlitz’s identities [6, eq. (3.22-3.23)]:
m m+r
B = B , (6)
n+m,r (j +r) n,r+j
j=0 r
X
m m+r
B = (−1)m−j B .
n,r+m "j +r# n+j,r
j=0 r
X
Here n is an r-Stirling number of the first kind (see [4,6,7]).
m r
h i
5 Dobinski’s formula
The Bell numbers are involved in Dobinski’s nice formula [9,13,14,19]:
1 ∞ kn
B = .
n
e k!
k=0
X
Our goal is to generalize this identity to our case.
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Theorem 5.1 (Dobinski’s formula) The r-Bell polynomials satisfy the i-
dentity
1 ∞ (k +r)n
B (x) = xk.
n,r ex k!
k=0
X
Consequently, the r-Bell numbers are generated by
1 ∞ (k +r)n
B = .
n,r
e k!
k=0
X
Proof. The r-Stirling numbers for a fixed n (and r) have the ’horizontal’ gen-
erating function [4]
n n+r
(x+r)n = xk,
(k +r)
k=0 r
X
whence, for an arbitrary integer m,
(m+r)n m n+r 1
= .
m! (k +r) (m−k)!
k=0 r
X
In the next step we multiply both sides with xm and sum from m = 0 to ∞.
Then
∞ (m+r)n ∞ m n+r xm n n+r
xm = = ex xk .
m! (k +r) (m−k)! (k +r) !
m=0 m=0k=0 r k=0 r
X X X X
2
We can determine some interesting sum with the aid of r-Bell numbers. For
example, we know from the second paragraph that B = 10, so
2,2
1 ∞ (k +2)2
= 10.
e k!
k=0
X
6 An integral representation
In 1885, Cesa`ro [8] found an amazing integral representation of the Bell num-
bers (see also [3,5]):
B = 2n! Im πeeeiθ sin(nθ)dθ.
n
πe
Z0
It is not hard to deduce the ’r-Bell version’.
Theorem 6.1 The r-Bell numbers have the integral representation as
B = 2n! Im πeeeiθereiθ sin(nθ)dθ.
n,r
πe
Z0
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Proof. In [6] we find that
n+r k k
k! = (−1)k−j (j +r)n. (7)
(k +r) j!
r j=0
X
In [5] the next equality appears:
π πjn
Im ejeiθ sin(nθ)dθ = . (8)
2 n!
Z0
Unifying the equations (7)-(8), we get that
π 1 n+r 1 k k π
= (−1)k−j Im e(j+r)eiθ sin(nθ)dθ
2 n!(k +r)r k! j=0 j! Z0
X
= 1 Im π k (−1)k−j k eeiθ j ereiθ sin(nθ)dθ
k! Z0 jX=0 j!(cid:16) (cid:17) 
π eeiθ −1 k 
=Im ereiθ sin(nθ)dθ,
(cid:16) k! (cid:17)
Z0
whence
∞ n+r 2n! π ∞ eeiθ −1 k
= Im ereiθ sin(nθ)dθ,
k=0(k +r)r π Z0 k=0 (cid:16) k! (cid:17) 
X X
 
 
2
and the result follows.
Remark 6.2 The imaginary part of the above integral can be calculated with
a bit of effort. The result is
2n! π
B = eecosθcossinθ+rcosθ·
n,r
πe
Z0
· cos(ecosθsinsinθ)sin(rsinθ)+sin(ecosθsinsinθ)cos(rsinθ) sin(nθ)dθ.
h i
Without the r-Bellnumbers in background, the evaluation of this integral seems
to be impossible...
Citing the general version of Dobinski’sformula we find the compellingidentity
∞ (k +r)n = 2n! Im πeeeiθereiθ sin(nθ)dθ.
k! π
k=0 Z0
X
9
7 Hankel transformation and log-convexity
Since
∞ tn
et B (x) = ex(et−1)+(r+1)t,
n,r
n!
n=0
X
Cauchy’s product immediately implies the next
Theorem 7.1 The r-Bell polynomials satisfy the relations
n n
B (x)= B (x),
n,r+1 k k,r
!
k=0
X
n n
B (x)= (−1)n−kB (x).
n,r k k,r+1
!
k=0
X
Aninteresting corollaryisconnected withthenotionof Hankel transform. The
H Hankel matrix [15] of an integer sequence (a ) is
n
a a a a ···
0 1 2 3
 
a a a a ···
1 2 3 4
H =  ,
 
a a a a ···
 2 3 4 5 
 ... ... ... ... ...
 
 
 
while the Hankel matrix of order n, denoted by h , is the upper-left submatrix
n
of H of size n × n. The Hankel transform of the sequence (a ) is again a
n
sequence formed by the determinants of the matrices h .
n
A notable result of Aigner and Lenard [2,16] is that the Hankel transform of
the sequence of Bell numbers is (1!,1!2!,1!2!3!,...), that is, for any fixed n,
B B B ··· B
0 1 2 n
(cid:12) (cid:12)
(cid:12)B B B ··· B (cid:12) n
(cid:12) 1 2 3 n+1(cid:12)
(cid:12) . . . . (cid:12) = i!
(cid:12) . . . . (cid:12)
(cid:12) . . . . (cid:12) i=0
(cid:12) (cid:12) Y
(cid:12) (cid:12)
(cid:12)B B B ··· B (cid:12)
(cid:12) n n+1 n+2 2n (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
We can determine the Hankel transform of r-Bell numbers easily. To reach
this aim, we recall an other notion. If (a ) is a sequence, then its binomial
n
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