Download Ces`aro`s Integral Formula for the Bell Numbers (Corrected)

Cesàro’s Integral Formula for the Bell Numbers (Corrected)
DAVID CALLAN
Department of Statistics
University of Wisconsin-Madison
Medical Science Center
1300 University Ave
Madison, WI 53706-1532
[email protected]
October 3, 2005
In 1885, Cesàro [1] gave the remarkable formula
Z π
2
cos θ
ee cos(sin θ)) sin( ecos θ sin(sin θ) ) sin pθ dθ
Np =
πe 0
where (Np )p≥1 = (1, 2, 5, 15, 52, 203, . . .) are the modern-day Bell numbers. This formula
was reproduced verbatim in the Editorial Comment on a 1941 Monthly problem [2] (the
notation Np for Bell number was still in use then). I have not seen it in recent works and,
while it’s not very profound, I think it deserves to be better known.
Unfortunately, it contains a typographical error: a factor of p! is omitted. The correct
formula, with n in place of p and using Bn for Bell number, is
Z
2 n! π ecos θ cos(sin θ))
Bn =
e
sin( ecos θ sin(sin θ) ) sin nθ dθ
πe 0
n ≥ 1.
eiθ
The integrand is the imaginary part of ee sin nθ, and so an equivalent formula is
Z π
iθ
2 n!
ee
Bn =
Im
e sin nθ dθ .
πe
0
(1)
The formula (1) is quite simple to prove modulo a few standard facts about set par titions. Recall that the Stirling partition number nk is the number of partitions of
P [n] = {1, 2, . . . , n} into k nonempty blocks and the Bell number Bn = nk=1 nk counts
all partitions of [n]. Thus k! nk counts ordered partitions of [n] into k blocks (the k!
factor serves to order the blocks) or, equivalently, counts surjective functions f from [n]
1
onto [k] (the jth block is f −1 (j)). Since the number of unrestricted functions from [n] to
[j] is j n , a classic application of the inclusion-exclusion principle yields
X
k
n
k−j k
j n.
=
(−1)
k!
j
k
j=0
(2)
The trig identity underlying Cesàro’s formula is nothing more than the orthogonality
of sines on [0, π]:
Z

π
sin mθ sin nθ dθ = 2
0
π
0
if m = n,
if m 6= n
for m, n nonnegative integers. Using the Taylor expansion ex =
formula eiθ = cos θ + i sin θ, it follows that
Z π
jn π
jeiθ
Im
e sin nθ dθ =
n! 2
0
xm
m≥0 m!
P
and DeMoivre’s
(3)
for integer j ≥ 0. Now we show that
Im
Z
π
0
for integer k ≥ 0 (of course,
Proof
n
k
iθ
ee − 1
k!
k
sin nθ dθ
!
1 n π
=
n! k 2
(4)
= 0 for n > k = 0 and for k > n).
The binomial theorem implies the left hand side is
Z π
k
1 X
jeiθ
k−j k
Im
e sin nθ dθ
(−1)
j
k! j=0
0
n
k
1 X
k−j k j π
(−1)
=
j n! 2
(3) k!
j=0
1 n π
=
(2) n! k 2
Finally, summing (4) over k ≥ 0 yields Cesàro’s formula (1). The Bell numbers have
many other pretty representations, including Dobinski’s infinite sum formula [3, p. 210]
∞
1 X kn
.
Bn =
e k=0 k!
2
References
[1] M. E. Cesàro, Sur une équation aux différences mêlées, Nouvelles Annales de Math.
(3), 4 (1885), 36–40.
[2] H. W. Becker and D. H. Browne, Problem E461 and solution, Amer. Math. Monthly
48 (1941), 701–703.
[3] L. Comtet, Advanced Combinatorics, D. Reidel, Boston, 1974.
AMS Classification numbers: 05A19, 05A15.
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