Articles of bernoulli numbers

An identity for Bernoulli numbers

Denote by $B_n$ the Bernoulli sequence (defined by the exponential generating function $\frac{x}{e^x-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n$). As we know $$ \sum^{n}_{j=0}\binom{n}{j}B_j=B_n\; ; \; n\geq 2 $$ what about $\sum^{n}_{j=0}(-1)^j\binom{n}{j}B_j$, and as a more general case $$ \sum^{n}_{j=0}\frac{(-1)^{j+1-k}}{j+1}\binom{n}{j}\binom{j+1}{k}B_{j+1-k} $$ where $k$ is a given integer such that $1\leq k\leq n+1$ (is there any similar identity?). Note that putting $k=1$ […]

How can I calculate closed form of a sum?

As we know the closed form of $$ {1 \over k} {n \choose k – 1} \sum_{j = 0}^{n + 1 – k}\left(-1\right)^{\, j} {n + 1 – k \choose j}B_{j} = \bbox[10px,border:2px solid #00A000]{{n! \over k!}\, \left[\, z^{n + 1 – k}\,\,\,\right] {z \exp\left(z\right) \over 1 – \exp\left(-z\right)}} $$ where $B_{n}$ the Bernoulli sequence […]

References mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers?

Is there anyone knows where is some official reference mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers (

Original author of an exponential generating function for the Bernoulli numbers?

The Bernoulli numbers were being used long before Bernoulli wrote about them, but according to Wikipedia, “The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants B0, B1, B2, … which provide a uniform formula for all sums of powers.” Did he publish an exponential generating […]

Power Series With Bernoulli Numbers

The exercise reads “Express the power series for $\large \frac{z}{\sin (z)} = \frac{2 i z}{e^{iz} – e^{-iz}} $ in terms of Bernoulli numbers.” I am given in a previous exercise that the Bernoulli numbers are defined by $$ \frac{z}{e^z – 1} = \sum_{n = 0}^{\infty} \frac{B_n}{n!} z^n, $$ where $ B_n $ is the $ […]

Relation between bernoulli number recursions

The recursion formula $$\sum_{k=0}^{n-1}{n\choose k}B_k=0$$ which is equation (34) from the MathWorld page is the most basic bernoulli number recursion. Another recursion is given by the following:Consider $$\mathbf{A} = \begin{bmatrix} -\frac{1}{2} & -\frac{1}{6} & -\frac{1}{12} & -\frac{1}{20} & -\frac{1}{30} & -\frac{1}{42}\\ \frac{1}{1} & 0 & 0 & 0 & 0 & 0\\ 0 & \frac{1}{2} […]

Bernoulli Numbers

I’ve read that Bernoulli Numbers are defined by the series $$ \frac{z}{e^z-1}\equiv \sum\limits_{n=0}^{\infty}B_n\frac{z^n}{n!},$$ So if I start with $0$ I get $$ B_0\frac{1}{1}=B_0{1}. $$ My question is, why is there a $B_0$ in the term…is it of any significance? Or just a “marker” or something to indicate that this is the $B_0$ term? If I […]

Is $\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$ a rational number for $m,n\ge 2\in\mathbb N$?

Question : Is $$\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$$ a rational number for $m,n\ge 2\in\mathbb N$ where $\zeta (s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$? Motivation : We know that $$\zeta (2k)=(-1)^{k+1}\frac{B_{2k}(2\pi)^{2k}}{2(2k)!}$$ and that $B_{2k}$ is a rational number for any $k\in\mathbb N$ where $B_n$ is the Bernoulli numbers. Hence, if both $m$ and $n$ are even, then we can see that $$\frac{\zeta […]

Continued fraction expansion related to exponential generating function

A recent SciComp.SE Question motivates us to ask for a nice continued fraction expansion of the following Maclaurin series: $$ f(x) = \sum_{n=0}^\infty \frac{B_n\; x^{n+3}}{n! (n+3)} = \int_0^x \frac{t^3}{e^t – 1} dt $$ where $B_n$ are the (first) Bernoulli numbers. Although the positive power of $t$ in the numerator of the integrand removes the singularity […]

Expected value of Bernoulli random variables

Consider a sequence of $n$ Bernoulli trials with $P(\text{success})=p$. Let $X_i$ and $X_j$ be indicator variables of the number of “success” in $i$th and $j$th runs. Given the total number of success was $m$, $m<n$. I am asked to compute the correlation coefficient for $X_i$ and $X_j$. Now I know the formula for correlation coefficient, […]