The factorially normalized Bernoulli polynomial $b_n(x) = B_n(x)/n!$ is known to be characterized by $b_0(x) = 1$ and $n >0$ $b_n(x)$ of is the anti-derivative of $b_{n-1}(x)$ according to $\int_0^1 b_n(x) dx = 0$. We provide related characterizations: $b_1(x) = x – 1/2$ and $(-1)^{n-1} b_n(x)$ for $n >0$ is $n$ times It’s a circular convolution of $b_1(x)$ with itself. Similarly, $1 – 2^n b_n(x)$ is the probability density at $x \in (0,1)$ of the fractional part of the sum of the $n$ independent random variables, each of which is beta$(1, 2) $ probability density $2(1-x)$ at $ x \in (0,1)$. The result has a new combination analogue, the {\em Bernoulli clock}. From the multiset $\{1,1, 2,2, \ldots, n, n\}$, or $2 n$ hours, two different times are uniformly randomized and marked as $1$, and the remaining From the $2 n – 2$ hours, pick two different uniformly random hours and mark them as $2$. upon. Start at time $0 = 2n$, move clockwise to the first time marked $1$, continue clockwise to the first time marked $2$, and so on marked $n$ The first of the 2 hours observed occurred at a random time $I_n$ between $1$ and $2n$. For each positive integer $n$, we say that 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)$ is the well-known combination of symmetries of the Bernoulli polynomial $b_n(1-x) = (-1)^n Analog. b_n(x)$.
