Questions tagged [bernoulli-numbers]

Questions on Bernoulli numbers, a special sequence of rational numbers that arise as the coefficients in the power series expansions of certain elementary functions.

292 questions
22 views

How to understand if an exercise is Poisson-Bernoulli or e -What are the difference

I have 3 different exercises and I am getting confused.If it is Poison or Bernoulli or e. I have seen and I know each and read, but I am not so clever to get it.I mean i can't detect the exercise what ...
29 views

Bernoulli processes and information entropy

) Given two iid Bernoulli processes B1 and B2. B1 is denoted by a sequence of random variables with Xi={0,1} and Pr{Xi=1} = 1/3 and B2 is denoted by a sequence of random variables with {Yi}={0, 1, 2} ...
61 views

Verifying this limit of a sum

A lenghty limit, $$\lim_{ n \to \infty}-\sqrt{8}\cdot \frac{n!}{B_n}\sum_{j=0}^{n}\frac{(1-2^{1-j})(1-2^{1+j-n})B_{n-j}B_j}{4^j(n-j)!j!}=\pi$$ The limit seems to appraoches to $\pi$ but I am not ...
68 views

Euler sum with Bernoulli numbers

In many sources, I find such equality: $$\dfrac{1}{n}\sum_{k=1}^n \binom n k B_kB_{n-k}+B_{n-1}=-B_n$$ where $B_1=-\dfrac{1}{2}$ However, there don't write how to get it. I think that it's ...
13 views

Recurrence for A000670

A000670 contain a formula by Martin Kochanski: Recurrence: $2a(n)=(a+1)^n$ where superscripts are converted to subscripts after binomial expansion - reminiscent of Bernoulli numbers $B_n=(B+1)^n$. ...
43 views

Identity of Bernoulli polynomials [duplicate]

I am trying to prove the following identity: $$B_n(1-x)=(-1)^nB_n(x)$$ I know that $$B_n(1-x)=\sum_{k=0}^n\binom n k B_n\cdot (1-x)^{n-k}$$ (I do not know how to write here exactly what I want. So I ...
97 views

Sum with Bernoulli numbers

How to prove that: $$\sum_{k=0}^n \binom n k 2^k B_k = (2-2^n)B_n$$ In this sum, $B_n$ is the Bernoulli number with $B_1 = -\frac 1 2$. Thanks for your attention!