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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. 3