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

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Functional Choice for p in a Bernoulli Distribution

Why is the functional choice $p = \exp(x)/(1+\exp(x))$ to model $p$ a good one in a Bernoulli distribution? Is it because it is limited at $0$ as $x$ approaches $0$ and $1$ as $x$ approaches ...
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1answer
34 views

{Probability}: choosing keys from a pool without replacement

The OP is trying to understand the following question. The OP understand that if you can always write out the term $$P(X=k) \implies (1-\frac{1}{N})(1-\frac{1}{N-1})\cdots(1-\frac{1}{N-k+1}),$$ ...
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1answer
51 views

Probability: deviation from the mean

I am having trouble to understand the following. If $S_n=X_1+X_2+......+X_n$, where X_1,X_2 are Bernouli (p). I don't understand this. So you get an intermediate point Constant* sqrt(n). To the ...
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1answer
33 views

Probability exercise Bernoulli. [closed]

Probability random signals. Im late I have no idea to start and this is for tomorrow. I was on training and have no break to do this work. I do this.You are an Internet savvy and enjoy watching video ...
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1answer
62 views

Question about $ \int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s) \forall s\in \Bbb N$

what I found from messing around was $$ \int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s) $$ $$ s\in \mathbb{N} $$ when the partial sum is changed to an equivalent polynomial using Faulhaber's formula. ...
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4answers
137 views

How is the Bernoulli numbers? For example, as against $B_2$?

How is the Bernoulli numbers? For example, as against $B_2$? For example, found that in internet $$\sum_{n=0}^{\infty} \frac{B_n\;x^n}{n!}=\frac{x}{e^x-1}$$ but if I want to find $B_2$ then ...
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1answer
24 views

dual formula to Bernoulli polynomials

$$ \tilde{B}_n(x) = \frac{(-1)^{n + 1}}{n!} \left( \delta^{(n - 1)}(x - 1) - \delta^{(n - 1)}(x) \right) $$ Wikipedia says this formulae is DUAL to the Bernoulli POlynomials but dual in what sense ?? ...
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0answers
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Showing that Faulhaber's formula is a polynomial of degree $m+1$ [duplicate]

The Faulhaber polynomials $S_{m}(n)$ are defined as $$S_{m}(n)=\sum_{k=1}^{n}k^{m}$$ It is the case that $S_{m}(n)$ is a polynomial in $n$ of degree $m+1$; I've seen a proof of this before, but I ...
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1answer
40 views

Two-term Binomial-Bernoulli Transform

The binomial transform states that if one has two real sequences $\{a_k \}$ and $\{b_k \}$ satisfying $b_n = \sum_{k = 0}^{n} \binom{n}{k} a_k$, then $a_n = \sum_{k = 0}^{n} (-1)^{n-k} \binom{n}{k} ...
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0answers
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Find the closed form of $S(m, n)=\sum_{j=1}^{m}\sum_{i=1}^{n}i^j$

I know that the closed form of the sum $\sum\limits_{i=1}^{n}i^j$ can be written using Bernoulli numbers $B_k$. It is the famous Faulhaber's formula: $$\sum\limits_{i=1}^{n}i^j =\frac{1}{j+1} ...
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2answers
27 views

Finding a Correlation between Bernoulli Variables?

Let X and Y be Bernoulli random variables. We don't assume independence or identical distribution, but we do assume that all 4 of the following probabilities are nonzero. Let a := P[X = 1, Y = 1], b ...
2
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2answers
65 views

How does one derive this formula $\zeta(-n) =\frac{B_{n+1}}{n+1}$?

$$\zeta(-n) =-\frac{B_{n+1}}{n+1}$$ What is the motivation or the derivation of this formula?
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0answers
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Is there a closed form formula for the Bernoulli numbers?

A while ago I found this algorithm. Today I read in wikipedia that Euler zig zag numbers can be used for computing the Bernoulli numbers. This Mathematica program computes the Euler zig zag numbers ...
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1answer
238 views

Convergence of $\sum_\lambda \frac{1}{1-\lambda x}$ where $p(\lambda)=0$ for a certain polynomial $p$

The powers of the roots $\lambda$ of these polynomials $$p_n(x):=\sum_{k=1}^{n-1}\frac{n!}{(n-k)!}x^{k-1}$$ (compare with the $p_n$ here) sum to these values $$\sum_\lambda ...
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0answers
51 views

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 ...
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1answer
139 views

Roots of some modified Bernoulli polynomials

Update The polynomials are generated as follows: Where $B_n(x) = \sum_{k=0}^n {n \choose k} b_{n-k} x^k$ is used to generate standard Bernoulli polynomials, top plot is generated as follows: ...
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5answers
699 views

Generalizing the sum of consecutive cubes $\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$ to other odd powers

We have, $$\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$$ $$2\sum_{k=1}^n k^5 = -\Big(\sum_{k=1}^n k\Big)^2+3\Big(\sum_{k=1}^n k^2\Big)^2$$ $$2\sum_{k=1}^n k^7 = \Big(\sum_{k=1}^n ...
4
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1answer
128 views

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 ...
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2answers
91 views

Verifying a relation involving Bernoulli polynomials

I would appreciate help, please, as to how to verify this relation from Kato's "Fermat's Dream" p.96. He say: By the definition of $B_n(x)$, the Bernoulli polynomial, we have ...
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0answers
55 views

Probability of Runs of Heads of Length N [duplicate]

For example: $“THHTHTTHHHTHTHTTHHTHT”$ contains 1 run of heads of length 3, 2 runs of length 2, and 4 runs of length 1. Assuming $P(H) = p$ and $P(T) = (1-p)$, calculate (using properties such as ...
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3answers
114 views

Bernoulli numbers: comparison to factorials

I am trying to understand the behaviour of the Bernoulli numbers with respect to factorials, specifically I'd like to know whether it is true that, for all $n \in N$ with $n \ge 2$ we have $$ ...
3
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1answer
87 views

show that $\lim_{x\to \frac14} \frac{d^{2k-1}}{dx^{2k-1}}\cot(\pi x)=-(2\pi)^{2k-1}2^{2k}(2^{2k}-1) \frac{\left | B_{2k} \right |}{2k} $

I try to prove the relation between Polygamma function and Bernoulli numbers but I faced this problem,is how to show that $$\lim_{x\to \frac14} \frac{d^{2k-1}}{dx^{2k-1}}\cot(\pi ...
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0answers
67 views

Are there negative Bernoulli numbers?

If not, why not? Also I don't mean Bernoulli number that are negative such as $B_4 = \frac{-1}{30}$ but numbers like $B_{-4} = ?$
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0answers
66 views

Is there an explicit formula for the Bernoulli numbers that doesn't implicitly recapitulate Faulhaber's formula?

So, I'm familiar with the "standard" explicit formula for the Bernoulli numbers: $$B_m (n) = \sum^m_{k=0}\sum^k_{v=0}(-1)^v {k \choose v} {(n+v)^m \over k+1}$$ where choosing $n=0$ gives the ...
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0answers
28 views

Hypergeometric distribution, $Hg(1,a,b)$ follows Bernoulli with $Be(\frac{a}{a+b})$

The probability function of Hypergeometric distribution , $Hg(n,a,b)$ is $$P(X=m)=\frac{\binom{a}{m}\binom{b}{n-m}}{\binom{a+b}{n}}$$ I have to show $Hg(1,a,b)$ follows Bernoulli with ...
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1answer
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Bernoulli Trial variables?

You are given n=10,000 light bulbs. Each has a reliability of p=99.99% Suppose you select a batch of r=190 light bulbs to light your warehouse. Can you use the equation to predict the ...
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1answer
88 views

On the numerators of Bernoulli numbers

Von Staudt-Clausen theorem implies that $pB_{2n} \in \mathbb{Z}_{p}$ for all primes $p$ and for all $n \in \mathbb{N}$. It means that the highest power of any prime that can occur in the denominator ...
4
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1answer
155 views

Expressing an integral in terms of the Bernoulli numbers

In Ahlfors' Complex Analysis text, the Bernoulli numbers, $B_k$, are defined as the coefficients in a Laurent development: $$(e^z-1)^{-1}=\frac{1}{z}-\frac{1}{2}+ \sum_1^\infty (-1)^{k-1} ...
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2answers
107 views

how to prove that $\psi_1(x)=\sum_{n=0}^{\infty}\frac{B_n}{x^{n+1}}$ ?

how to prove that $$\psi_1(x)=\sum_{n=0}^{\infty}\frac{B_n}{x^{n+1}}$$ where $\psi_1(z)$ is Trigamma function and $B_n$ is Bernoulli number
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1answer
114 views

Infinite sum and equality between coefficients of the same index

I have two infinite sums that forms an equality: $$\sum_{n=1}^{\infty} \left(\zeta(2n)\frac{x^{2n}}{\pi^{2n}}\right) = \sum_{n=1}^\infty ...
7
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1answer
595 views

Bernoulli numbers generating function

Consider the following generating formula: $$\frac{t}{e^t-1}=\sum_{n=1}^{\infty} B_n \frac{t^n}{n!}$$ There is some intuitive explanation about it? I want to know because I need to proof to myself ...
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3answers
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The vanishing of the $\bf B$s

Define $B_0=1$ and recursively $$\tag 1 \sum_{k=0}^{n}\binom {n+1}k B_k=[n=0]$$ How can I prove $B_{2n+1}=0$ for $n\geqslant 1$ using this definition? Note the above means that ...
0
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1answer
165 views

Probability - rolling a fair die 10 times, what is the probability you would match a separate set of 10 numbers?

Having some trouble with this problem... Say someone is rolling a fair die 10 times, and using that roll as an attempt to guess what number (1-6) someone else has written down on a piece of paper for ...
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2answers
805 views

Explicit formula for Bernoulli numbers by using only the recurrence relation

It is not hard to show, by induction on $m\in\mathbb N$, that there exist a sequence $(B_n)_{n\geq0}$ of rational numbers such that ...
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0answers
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Sum of power and Bernoulli intuitive discover

I really looked up to the 10th page of google, and the PDF's I find aren't complete, and definetively have NOT intuitive explanation about Bernoulli's discover. I know that he observed the formulas of ...
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4answers
110 views

Telescoping sum of powers

$$ \begin{array}{rclll} n^3-(n-1)^3 &= &3n^2 &-3n &+1\\ (n-1)^3-(n-2)^3 &= &3(n-1)^2 &-3(n-1) &+1\\ (n-2)^3-(n-3)^3 &= &3(n-2)^2 &-3(n-2) &+1\\ ...
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1answer
156 views

Proving the Bernoulli number relation $(1+B)^n=B^n$

We know that we can generate the Bernoulli numbers using the relation $(1+B)^n=B^{[n]}$ where $B_n$ is $n$th Bernoulli number. But how we can prove this works? Thanks to all. Edit 2: is there a ...
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2answers
145 views

Radius of convergence of the Bernoulli polynomial generating function power series.

The generating function of the Bernoulli Polynomials is: $$\frac{te^{xt}}{e^t-1}=\sum_{k=0}^\infty B_k(x)\frac{t^k}{k!}.$$ Would it be right to say that the radius of convergence of this power series ...
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1answer
244 views

Radius of convergence of a power series with Bernoulli numbers

Say, we use the definition: Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$ and then derive power series representations of the ...
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0answers
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Bernoulli formula

The sum: $$S_m(n) = 1^m + 2^m + 3^m + 4^m + 5^m...+ n^m$$ Can be calculated by this formula, called the "Bernoulli formula" in wikipedia $$S_m(n) = \frac{1}{m+1}\sum_{k=0}^m {m+1\choose k}B_k ...
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5answers
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When are we (not) allowed to replace $x$ by $ix$?

It seems to be quite a common manipulation to replace $x$ by $ix$. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...
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2answers
82 views

How does one get the Bernoulli numbers via the generating function?

Here is the definition: Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$ I've tried to naively expand $\frac{x}{e^x-1}$ around ...
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1answer
50 views

Intuitive bernoulli numbers

Can somebody explain me or give me a link with a intuitive point of view of Bernoulli numbers? I mean, somebody just saw a typical sequence of numbers that appears in some taylor expansions, and them ...
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1answer
155 views

A continued fraction involving Bernoulli numbers

Let $B_n$ be the Bernoulli numbers. Then, we can write a function $F(x)$, expressed as a continued fraction involving those $B_n$, such that it gives the form, $$ \displaystyle \displaystyle ...
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1answer
105 views

how to prove binomial through bernoulli indicators ??

how to prove binomial through Bernoulli indicators? is it x-bernoulli (P) y=x1,x2,...,xn. where xi is the independent variable bernoulli gives y-Bin(n,p)?
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2answers
334 views

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 ...
2
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0answers
64 views

Bernoulli generating function and cotangent

May I ask for a little help in solving a problem about Bernoulli number generating function? Bernoulli number generating function is given by: $$f(z):=\begin{cases} \frac{z}{e^{z}-1} & z \in ...
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1answer
210 views

Bernoulli polynomials properties

I was reading about Bernoulli polynomials in this article: http://ocw.mit.edu/courses/mathematics/18-100c-analysis-i-spring-2006/projects/silva.pdf and I saw this property: $$B_n(1−x) = ...
3
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1answer
159 views

Complex Taylor series and Bernoulli numbers

Let: $f(z)=\frac{z}{e^z-1}$ if $z\ne0$, and $f(z)=1$ if $z=0$. Please help to prove that $\sum_{k=0}^{n-1}\binom n kf^{(k)}(0)=0$ for any $n>1$ and $f^{(2n+1)}(0)=0$ for any natrual $n$.
3
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324 views

Proof for generalized sum of powers

Bernouli's Formula for sum of kth powers of first n natural numbers is given by: $$f_k(n)=\frac{1}{k+1}\sum_{j=0}^k{k+1\choose j}B_j(n+1)^{k+1-j}$$ where $Bj$ is the $j^{th}$ Bernoulli Number and is ...