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Ces`Aro's Integral Formula for the Bell Numbers (Corrected)

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Ces`Aro's Integral Formula for the Bell Numbers (Corrected) Ces`aro’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, Ces`aro [1] gave the remarkable formula π 2 cos θ N = ee cos(sin θ)) sin( ecos θ sin(sin θ) ) sin pθ dθ p πe Z0 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 π 2 n! cos θ B = ee cos(sin θ)) sin( ecos θ sin(sin θ) ) sin nθ dθ n ≥ 1. n πe Z0 eiθ The integrand is the imaginary part of ee sin nθ, and so an equivalent formula is π 2 n! eiθ B = Im ee sin nθ dθ . (1) n πe Z0 The formula (1) is quite simple to prove modulo a few standard facts about set par- n titions. Recall that the Stirling partition number k is the number of partitions of n n [n] = {1, 2,...,n} into k nonempty blocks and the Bell number Bn = k=1 k counts n k n n k k all partitions of [ ]. Thus ! k counts ordered partitions of [ ] into P blocks (the ! 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 jn, a classic application of the inclusion-exclusion principle yields n k k k! = (−1)k−j jn. (2) k j j=0 X The trig identity underlying Ces`aro’s formula is nothing more than the orthogonality of sines on [0, π]: π π if m = n, sin mθ sin nθ dθ = 2 0 m n Z 0 if =6 m m, n ex x for nonnegative integers. Using the Taylor expansion = m≥0 m! and DeMoivre’s iθ formula e = cos θ + i sin θ, it follows that P π n iθ j π Im eje sin nθ dθ = (3) n! 2 Z0 for integer j ≥ 0. Now we show that iθ k π ee − 1 1 n π Im sin nθ dθ = (4) k! n! k 2 Z0 ! n for integer k ≥ 0 (of course, k = 0 for n>k = 0 and for k > n). Proof The binomial theorem implies the left hand side is k π 1 k iθ (−1)k−j Im eje sin nθ dθ k! j j=0 0 X Z k 1 k jn π = (−1)k−j ( ) k! j n! 2 3 j=0 X 1 n π = n! k 2 (2) Finally, summing (4) over k ≥ 0 yields Ces`aro’s formula (1). The Bell numbers have many other pretty representations, including Dobinski’s infinite sum formula [3, p. 210] ∞ 1 kn B = . n e k! k X=0 2 References [1] M. E. Ces`aro, Sur une ´equation aux diff´erences mˆel´ees, 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. 3.
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