The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution - Yassine El Maazouz; Pitman, Jim (arXiv.org, 2024)
The factorially normalized Bernoulli polynomials \(b_n(x) = B_n(x)/n!\) are known to be characterized by \(b_0(x) = 1\) and \(b_n(x)\) for \(n >0\) is the antiderivative of \(b_{n-1}(x)\) subject to \(\int_0^1 b_n(x) dx = 0\). We offer a related characterization: \(b_1(x) = x - 1/2\) and \((-1)^{n-1} b_n(x)\) for \(n >0\) is the \(n\)-fold circular convolution of \(b_1(x)\) with itself. Equivalently, \(1 - 2^n b_n(x)\) is the probability density at \(x \in (0,1)\) of the fractional part of a sum of \(n\) independent random variables, each with the beta\((1,2)\) probability density \(2(1-x)\) at \(x \in (0,1)\). This result has a novel combinatorial analog, the {\em Bernoulli clock}: mark the hours of a \(2 n\) hour clock by a uniform random permutation of the multiset \(\{1,1, 2,2, \ldots, n,n\}\), meaning pick two different hours uniformly at random from the \(2 n\) hours and mark them \(1\), then pick two different hours uniformly at random from the remaining \(2 n - 2\) hours and mark them \(2\), and so on. Starting from hour \(0 = 2n\), move clockwise to the first hour marked \(1\), continue clockwise to the first hour marked \(2\), and so on, continuing clockwise around the Bernoulli clock until the first of the two hours marked \(n\) is encountered, at a random hour \(I_n\) between \(1\) and \(2n\). We show that for each positive integer \(n\), the event \(( I_n = 1)\) has probability \((1 - 2^n b_n(0))/(2n)\), where \(n! b_n(0) = B_n(0)\) is the \(n\)th Bernoulli number. For \( 1 \le k \le 2 n\), the difference \(\delta_n(k):= 1/(2n) - \P( I_n = k)\) is a polynomial function of \(k\) with the surprising symmetry \(\delta_n( 2 n + 1 - k) = (-1)^n \delta_n(k)\), which is a combinatorial analog of the well known symmetry of Bernoulli polynomials \(b_n(1-x) = (-1)^n b_n(x)\).