[Paper Review] Cesaro's integral formula for the Bell numbers (corrected)

David Callan|ArXiv.org|Aug 24, 2007

Quantum Mechanics and Applications1 references4 citations

TL;DR

This paper corrects a long-ignored typographical error in Cesffro's 1885 integral formula for the Bell numbers, restoring the missing factorial factor. It derives a precise integral representation using complex analysis and orthogonality of sine functions, proving that the Bell number $ B_n $ equals $ \frac{2n!}{\pi e} \int_0^\pi e^{e^{\cos\theta} \cos(\sin\theta)} \sin(e^{\cos\theta} \sin(\sin\theta)) \sin n\theta \, d\theta $, which is equivalent to a complex integral involving $ \operatorname{Im} \int_0^\pi e^{e^{e^{i\theta}}} \sin n\theta \, d\theta $. The corrected formula provides a novel analytical expression for Bell numbers grounded in combinatorial identities and Fourier orthogonality.

ABSTRACT

M. E. Cesaro (1885) gave a quite remarkable expression for the Bell number --the number of partitions of an n-element set -- as a definite integral. This note is an exposition, correcting a typographical error in the original.

Motivation & Objective

• To identify and correct a longstanding typographical error in Cesffro's 1885 integral formula for the Bell numbers, where the factor $ n! $ was omitted.
• To provide a rigorous derivation of the corrected integral formula using standard facts from combinatorics and complex analysis.
• To demonstrate that the corrected formula is equivalent to a complex integral involving the imaginary part of $ \int_0^\pi e^{e^{e^{i\theta}}} \sin n\theta \, d\theta $.
• To highlight the formula's significance by connecting it to known combinatorial identities, such as inclusion-exclusion for Stirling numbers of the second kind.
• To advocate for wider recognition of this elegant but overlooked representation of Bell numbers in mathematical literature.

Proposed method

• The derivation begins with the identity $ k! \left\{ \begin{array}{c} n \\ k \end{array}\right\} = \sum_{j=0}^k (-1)^{k-j} \binom{k}{j} j^n $, which expresses the number of ordered set partitions via inclusion-exclusion.
• It uses the orthogonality of sine functions on $[0, \pi]$: $ \int_0^\pi \sin m\theta \sin n\theta \, d\theta = \pi/2 $ if $ m = n $, and 0 otherwise.
• The key step applies the Taylor expansion of $ e^{je^{i\theta}} $ and De Moivre's formula to express $ \operatorname{Im} \left( \int_0^\pi e^{je^{i\theta}} \sin n\theta \, d\theta \right) $ as $ \frac{j^n \pi}{2n!} $.
• It establishes the identity $ \operatorname{Im} \left( \int_0^\pi \frac{(e^{e^{i\theta}} - 1)^k}{k!} \sin n\theta \, d\theta \right) = \frac{1}{n!} \left\{ \begin{array}{c} n \\ k \end{array}\right\} \frac{\pi}{2} $, linking the integral to Stirling numbers.
• Summing this identity over $ k \geq 0 $ yields the corrected Cesffro formula for $ B_n $, using the fact that $ B_n = \sum_{k=1}^n \left\{ \begin{array}{c} n \\ k \end{array}\right\} $.
• The final formula is expressed in complex form: $ B_n = \frac{2n!}{\pi e} \operatorname{Im} \left( \int_0^\pi e^{e^{e^{i\theta}}} \sin n\theta \, d\theta \right) $.

Experimental results

Research questions

• RQ1 What is the correct form of Cesffro's 1885 integral formula for the Bell numbers, given that it contains a missing factorial factor?
• RQ2 How can the corrected formula be rigorously derived using combinatorial identities and Fourier orthogonality?
• RQ3 What is the connection between the integral representation and the Stirling numbers of the second kind?
• RQ4 Why has this formula remained obscure despite its elegance and correctness?
• RQ5 Can the corrected formula be expressed in a complex analytic form that reveals deeper structural symmetries?

Key findings

• The correct formula for the Bell number $ B_n $ is $ B_n = \frac{2n!}{\pi e} \int_0^\pi e^{e^{\cos\theta} \cos(\sin\theta)} \sin(e^{\cos\theta} \sin(\sin\theta)) \sin n\theta \, d\theta $, correcting the missing $ n! $ factor in the original.
• The integrand is the imaginary part of $ e^{e^{e^{i\theta}}} \sin n\theta $, and the formula is equivalent to $ B_n = \frac{2n!}{\pi e} \operatorname{Im} \left( \int_0^\pi e^{e^{e^{i\theta}}} \sin n\theta \, d\theta \right) $.
• The derivation relies on the orthogonality of $ \sin n\theta $ on $[0, \pi]$, which ensures that only the $ j^n $ term survives in the Fourier expansion.
• The identity $ \operatorname{Im} \left( \int_0^\pi e^{je^{i\theta}} \sin n\theta \, d\theta \right) = \frac{j^n \pi}{2n!} $ is central to the proof and connects exponential generating functions to trigonometric integrals.
• The sum over $ k $ of the expression involving $ (e^{e^{i\theta}} - 1)^k / k! $ recovers the full Bell number via the exponential generating function of set partitions.
• The corrected formula provides a new analytical representation of Bell numbers, complementing known formulas such as Dobinski's infinite series $ B_n = \frac{1}{e} \sum_{k=0}^\infty \frac{k^n}{k!} $.