The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution

TL;DR

The paper introduces a novel circular-convolution framework for Bernoulli polynomials, showing that $b_n(x)=(-1)^{n-1} b_1^{\varoast n}(x)$ with $b_1(x)=x-1/2$, and connects this to a probabilistic model where $1-2^n b_n(u)$ is the density of the fractional part of a sum of $n$ Beta$(1,2)$ variables. It then develops the Bernoulli clock, a combinatorial permutation model on a $2n$-hour clock, establishing explicit distributions for the index $I_n$ and the number of turns $D_n$, and links these to Bernstein expansions and permutation counts. The work further generalizes to multi-multiset clocks, analyzes renewal processes with Beta jumps, and derives asymptotic laws and mean functions via exponential polynomials, thereby bridging Bernoulli polynomials, probability, and combinatorics. The wrapping-gamma framework in the final section embeds Bernoulli polynomials in the probabilistic analysis of circular distributions, offering a unified view of expansions on the circle through circular convolution and discrete-continuous analogies. Overall, the paper provides new mathematical connections between Bernoulli polynomials, circular convolution, random permutations, and renewal theory, with implications for both theory and combinatorial probability.

Abstract

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 $δ_n(k):= 1/(2n) - ¶( I_n = k)$ is a polynomial function of $k$ with the surprising symmetry $δ_n( 2 n + 1 - k) = (-1)^n δ_n(k)$, which is a combinatorial analog of the well known symmetry of Bernoulli polynomials $b_n(1-x) = (-1)^n b_n(x)$.

Figures (3)

• Figure 1: The clock is a circle of circumference $1$. Inside the circle the numbers $1,2, \ldots, 8$ index the order statistics of $8$ uniformly distributed random points on the circle. The corresponding numbers outside the circle are a random assignment of labels from the multiset of four pairs $1^2 2^2 3^2 4^2$. The four successive arrows delimit segments of $\mathbb{T} \equiv [0,1)$ whose lengths $X_1,X_2,X_3,X_4$ are independent beta$(1,2)$ random variables, while $(I_1,I_2,I_3,I_4)$ is the sequence of indices inside circle, at the end points of these four arrows. In this example, $(I_1,I_2,I_3,I_4) = (1,4,6,3)$, and the number of turns around the circle is $D_4 = 1$.
• Figure 2: Plots of $2n \pi^n \delta_n$ (dotted curve in blue), $(2\pi)^n b_n(x)$ (curve in red) and their difference (dotted curve in black) for $n = 70, 75, 80, 85$.
• Figure 3: Plots of $2n \pi^n \delta_{k:2n} - (2\pi)^n b_n \left( \frac{k-1}{2n -1 }\right)$ for $n = 100, 200, 300, 400, 500, 600$.

Theorems & Definitions (58)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3: Convolving monomials
• proof
• Remark 2.4
• Corollary 3.1