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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.

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Show that $\sum_{k=1}^n{2^{2k-1}\binom{2n+1}{2k}B_{2k}(0)}=n$

Lately, I've been working on a proof (whose context is not necessary to discuss) and I only need one last thing in order to finish it. To be more specific, for completeness it would suffice to show ...
Vaskara_GRek_O's user avatar
4 votes
1 answer
100 views

Sum of reciprocal Bernoulli numbers

What is sum of the Bernoulli numbers? discusses the sum of the Bernoulli numbers, using divergent sum methods since the Bernoulli numbers grow exponentially. This exponential growth makes it so that ...
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How the modified Bernoulli numbers relate to the ordinary Bernoulli numbers

The modified Bernoulli numbers are defined as the numbers $b_k$ whose generating series is $$\frac 1 2\log\left(\frac{\sinh \frac t 2}{\frac t 2}\right) = \sum_k b_k t^k.$$ (I use a slightly different ...
red_trumpet's user avatar
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About the $n$th derivative of the Riemann zeta function on positive even integers

I know there exist a formula for the Riemann zeta function on positive even integers involving Bernoulli numbers. Do there exist any closed form for the $n$th derivative of the Riemann zeta function ...
Haidara's user avatar
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Divergence of the Riemann Zeta function

Consider the equation $\displaystyle\zeta\left(s\right) \Gamma\left(s\right) = \int_{0}^{\infty}\frac{u^{s - 1}}{{\rm e}^{u} - 1} {\rm d}u$. This integral equals $\displaystyle\int_{0}^{\infty} \frac{...
Aspirant29's user avatar
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Literature check for a summation formula

In this paper I did a lot of stuff relating to sums, including a formula for explicitly evaluating them near the end of the paper: Let $f(x)=\sum_{k=0}^\infty c_kx^k$ be the power series of $f(x)$. ...
Kamal Saleh's user avatar
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Analytic continuation of Bernoulli numbers: $B_{1/2}$?

$B_0$ is 1, $B_1$ is $\pm 1/2$, and so on (I'm not going to list every single Bernoulli number). The Bernoulli numbers $B_n$ are defined for every integer $n \ge 0$. But what about $B_{1/2}$? And $B_i$...
Alexandra's user avatar
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$\zeta(s)=\frac{(-1)^{1-s}B_{1-s}}{s-1}$ by Ramanujan's Master Theorem

Let $f(x)=\frac{x}{e^x-1}=\sum_{k=0}^\infty \frac{(-1)^k B_k}{k!}(-x)^k$. It is well-kown that $$\int_0^\infty x^{s-2}f(x)dx=\Gamma(s)\zeta(s).\tag 1$$ On the other hand by Ramanujan's Master Theorem, ...
Bob Dobbs's user avatar
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Is there a closed sum to the following series?

Came across another interesting sum when trying to use Euler-Maclaurin on the ratio of a geometric series. Does anyone recognize a closed form to the following?: $$ \sum_{z=1}\frac{B_{2z}\ x^{2z}}{(2z)...
user3108815's user avatar
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The efficient computation of the zeta function at even integers

It is commonplace (see for instance this paper by McGown) to calculate the Riemann $ζ$ function at even integer values by using an approximation to the real value and the Staudt--von Clausen theorem. ...
Cloudscape's user avatar
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Value of the Riemann Zeta Function for negative integers

I am reading the book "Riemann's Zeta Function" by H. M. Edwards. I had a confusion in Section 1.5, Page 12 at the derivation of the formula of Riemann Zeta Function at negative integers i.e....
Souparna's user avatar
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A sum involving Bernoulli numbers and product of binomial coefficients

I came across a sum involving Bernoulli numbers and product of binomial coefficients during a computation. Precisely, it is $\sum\limits_{c=0}^m B_c\binom{r}{c}\binom{r-c}{m-c}$ where $B_c$ denotes ...
Vishnu Namboothiri K's user avatar
3 votes
2 answers
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Summation $\sum_{k=0}^{\lfloor\frac{n-1}{2}\rfloor}\frac{1}{\alpha^{2k+1}}\frac{B_{2k+2}}{\left(n-2k\right)!\left(2k+2\right)!}$

For $\alpha, n \in \text{N}$ How do we Prove That, $$\sum_{k=0}^{\lfloor\frac{n-1}{2}\rfloor}\frac{1}{\alpha^{2k+1}}\frac{B_{2k+2}}{\left(n-2k\right)!\left(2k+2\right)!}=\frac{\alpha}{n!\left(n+2\...
Miracle Invoker's user avatar
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3 answers
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Search for a geometrical way to calculate the summations $\sum_{k=1}^n k^p$ ($p=1,2,3...$)

There exist geometrical proofs for the calculation of the sums $S_p(n):=\sum_{k=1}^nk^p$ for $p=1$ and $p=2$ (e.g. compilation of proofs for sum of squares and cubes, understanding some PWoWs). The ...
feli_x's user avatar
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Maclaurin series without Bernoulli or Euler numbers?

I've been working through these listed Maclaurin series and deriving all of them. For the listed trigonometric functions, I'm wondering if there are ways to do things: not have Bernoulli or Euler ...
Ally's user avatar
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1 answer
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Closed form for $\sum_{n=1}^{\infty}\frac{(2\log\phi)^{2n+3}B_{2n}}{2n(2n+3)!}$

I need a closed form for the sum $$\sum_{n=1}^{\infty}\frac{(2\log\phi)^{2n+3}B_{2n}}{2n(2n+3)!} $$ where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio and $B_n$ are the Bernoulli numbers. I tried ...
Max's user avatar
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$p$-divisibility of consecutive Bernoulli numbers

Let $B_n$ be $n$-th Bernoulli number. We will write $p \mid B_n$ if prime $p$ divides the numerator of $B_n$. Now $p = 583$, we can show $p \mid B_{90}$ and $p \mid B_{92}$ (I found it using computer)....
Offlaw's user avatar
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Summation involving Bernoulli's numbers and Faulhaber's Formula

While manipulating the expression, $$n! = n(n-1)(n-2)\cdot\ldots\cdot 3\cdot 2\cdot 1, $$ (I think it would be better not to tell how I manipulated because it might be more confusing) I ended up ...
Rahul's user avatar
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1 answer
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What is the simple dependence of the diagonals (or columns) of the Faulhaber matrix on the first entry (Bernoulli numbers)?

In this presentation by Mathologer you can find the following slide: relating the sum of the $n$ first $k$ powers of integers, i.e. $S_k$ and the Faulhaber matrix diagonals (which correspond to the ...
JAP's user avatar
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An integer triangle for representing the Bernoulli numbers

We introduce a family of rational polynomials, for integer $n \ge 0$, as $$ r_{n}(x) = \sum_{k=0}^n \frac{\sum_{j=0}^k x^j \binom{k}{j}(j+1)^n}{k+1}. $$ Let $\operatorname{B}_n(x)$ denote the ...
Peter Luschny's user avatar
4 votes
1 answer
2k views

How to solve the Bernoulli Integral: $\int_0^1 x^xdx$ without a calculator?

So I was browsing the homepage of Youtube to see if there were any math equations that I thought that I might be able to solve when I came across this video by Dr. Trevor Bazett saying that the ...
CrSb0001's user avatar
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7 votes
2 answers
448 views

Is this a new representation of (some) Bernoulli numbers?

Let $\operatorname{B}(n)$ denote the Bernoulli numbers and $\operatorname{b}(n) = \operatorname{B}(n)/n$ with $b(0)=1$ the divided Bernoulli numbers. Also let $\sigma_{k}(n)= \sum_{d \mid n} d^k$ ...
Peter Luschny's user avatar
1 vote
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117 views

New asymptotic expression of $\zeta^{(m)}(1-s)$ involving generalized Stirling numbers of the $1$st kind

Introduction A few years ago I calculated this: For $x\to\infty$ $$\sum_{n=1}^{x}n^{s-1}\ln(n)^m\propto (-1)^m\left[\color{blue}{\zeta^{(m)}(1-s)}+\frac{1}{s}\sum_{j=0}^{s}\binom{s}{j}\ B_{s-j}^{+}x^j ...
Math Attack's user avatar
3 votes
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111 views

Value of $\pi$ and algorithm for Bernoulli numbers

Chowla and Hartung provide an "algorithm" for computing Bernoulli numbers in this paper (https://www.semanticscholar.org/paper/An-%22exact%22-formula-for-the-m-th-Bernoulli-number-Chowla-...
japjap's user avatar
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Given $x$ find minimum positive $N$ for which the following is composite $ 1^x+2^x+3^x+4^x+\cdots+N^x $

Problem: Given positive integer $x$ find minimum value of positive integer $N$ starting which the following is always composite $$ 1^x+2^x+3^x+4^x+\cdots+N^x $$ My Thoughts: This is a followup to this ...
sibillalazzerini's user avatar
1 vote
1 answer
91 views

Getting Bernoulli numbers via generating functions

In the below post, in the first answer(by "Start wearing purple" user) How does one get the Bernoulli numbers via the generating function? $\begin{align}\frac{1}{1+\frac{x}{2!}+\frac{x^2}{3!}...
shanrrg's user avatar
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5 votes
4 answers
235 views

Evaluation of $\int^{\infty}_{-\infty} dx \frac{x^2\cdot\exp\left(-x\right)}{\left(1+\exp\left(-x\right)\right)^2}$

I can see that Wolfram Alpha can compute this, but I would like to understand how. The integrand has a poles at $\pm i\pi$ but because of the square at the bottom of the fraction, I cannot quite see ...
Cryo's user avatar
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0 answers
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Integral form of Nørlund Polynomial

Nørlund Polynomials $B_n^s(0) $ are defined as $B_n^s(0)= \lim\limits_{t \rightarrow 0} \frac {d^n}{dt^n} \left ( \frac {t}{e^t-1} \right) ^s$ For $s=1$, $B_n^s(0)= 2\pi \int_{-\infty}^{\infty} \...
Wreior's user avatar
  • 396
2 votes
1 answer
67 views

Variance and mean of a bernoulli [closed]

Let ${X_i}$ be a sequence of iid bernoulli random variables sucht that $P\{X_i = 1\} = p$ and $P\{X_i = 0\} = 1-p$ and let $S_n = \sum_{i=1}^n X_i$. Calculate variance and mean of $\frac{S_n}{n}$ I'm ...
Ricter's user avatar
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0 answers
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Closed form expression for the sum of $i^a (n-i)^b$ from $0$ to $n$

Let $a,b\in\mathbb{N}$. It's well known that we have $$ \int_0^v x^a (v-x)^b\,dx = \frac{a!\, b!}{(a+b+1)!} v^{a+b+1} = \frac{v^{a+b+1}}{(a+b+1){a+b \choose a}} $$ There's also closed sum expression ...
Gro-Tsen's user avatar
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0 votes
1 answer
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Query on the inverse function of Faulhaber's formula

Even though the irregularities lying behind the Bernoulli's numbers are yet to be clarified, it seems that we already have algorithms to effectively compute them. [Source1, Source2] In this question, ...
user1851281's user avatar
0 votes
3 answers
200 views

$n$-th derivative of $1/(e^x - 1)$ (related with Bernoulli numbers)

It's well known that $$\frac{1}{e^x-1}=\sum_{n=0}^{\infty}B_n\frac{x^{n-1}}{n!}\hspace{1cm}\left(\text{or}\hspace{0.5cm} \frac{x}{e^x-1}=\sum_{n=0}^{\infty}B_n\frac{x^{n}}{n!}\right)$$ where $B_n$ are ...
popi's user avatar
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1 vote
0 answers
50 views

Is there an accurate representation of Bernoulli umbra?

Bernoulli umbra is some object $B$, an element of a commutative ring, such that there is an “index lowering” linear operator $\operatorname{eval}$ which applied to $B^n$ will give $B_n$, the $n$-th ...
Anixx's user avatar
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1 vote
0 answers
66 views

Why generating function of Bernoulli Numbers is not continuous?

When deriving the exponential generating function of Bernoulli Numbers and its closed form, I've stumbled upon a huge, to my rigorous heart, problem - discontinuity of the closed form. $$ B(t) = \...
Capy Maths's user avatar
17 votes
4 answers
647 views

Bernoulli numbers identity:$\sum_{k=0}^n\sum_{l=0}^m\binom{n}{k}\binom{m}{l}\frac{(n-k)!(m-l)!}{(n+m-k-l+1)!}(-1)^l B_{k+l}=0$,for all $n\ge1$,$m\ge0$

For all $n\geq 1$ and $m\geq0$, I'm trying to prove that $\sum_{k=0}^n\sum_{l=0}^m\binom{n}{k}\binom{m}{l}\frac{(n-k)!(m-l)!}{(n+m-k-l+1)!}(-1)^l B_{k+l}=0$ where $B_n$ are the Bernoulli numbers with $...
Sebastian's user avatar
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7 votes
4 answers
413 views

Bernoulli numbers alternate signs

Michael Spivak in Calculus, Fourth ed., Chapter 27, Problem 16 (page 572) defines the Bernoulli numbers based on $$ \frac{z}{e^z-1} = \sum_{n=0}^\infty\frac{B_nz^n}{n!}. $$ He asks the reader to ...
Richard Hevener's user avatar
1 vote
1 answer
663 views

How do you apply the digit-extraction algorithm from Plouffe (2022)?

I recently became interested in programs that calculate pi, and I was reading about various spigot algorithms when I came across this Wolfram MathWorld site: https://mathworld.wolfram.com/Digit-...
Denver M's user avatar
6 votes
1 answer
214 views

The relation of the Bernoulli numbers to the Catalan numbers

The Bernoulli numbers $B_n$ are the backbone of calculus, and according to B. Mazur, they "act as a unifying force, holding together seemingly disparate fields of mathematics." The Catalan ...
Peter Luschny's user avatar
0 votes
0 answers
64 views

Why does the asymptotic approximating function for a series sum of a harmonic progression diverge as Bernoulli terms are expanded?

From an earlier post, I learned that a closed form approximating expression for the series sum of a harmonic progression can be defined as $$\sum_{k=1}^{f}\frac{1}{1+ak}=\frac{1}{a} (H_{f+ \frac{1}{a} ...
Scott's user avatar
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1 vote
1 answer
104 views

Some questions about the values of Riemann zeta function at positive integers.

The Riemann zeta function can be defined by the counter integral (25) in herė: https://mathworld.wolfram.com/RiemannZetaFunction.html Now, by using the Maclaurin series of$\frac{z}{e^z-1}$ in (25), I ...
Bob Dobbs's user avatar
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2 votes
2 answers
77 views

Prove a Bernoulli polynomial equation: $B_s(x+y)=\sum_{j=0}^s \binom{s}{j} B_{j}(x)\cdot y^{s-j}$

I want to show the equation for Bernoulli polynomials $B_s(x+y)$: $$B_s(x+y)=\sum_{j=0}^s \binom{s}{j} B_{j}(x)\cdot y^{s-j}$$ I begin with: $$B_s(z)=\sum_{j=0}^s \binom{s}{j} B_{s-j} \cdot z^j$$ ...
MathFail's user avatar
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0 votes
1 answer
455 views

Euler Numbers and Bernoulli Numbers

Euler numbers, $E_{n}$, $n∈ℕ,$ are integers defined by the Maclaurin series $\frac{1}{\cosh x}=∑_{n=0}^{∞}\frac{E_{n}}{n!}x^n.$ Euler polynomials, $E_{n}(x), n∈ℕ,$ are defined by the generating ...
Bob Dobbs's user avatar
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3 votes
1 answer
92 views

Why is $B_{2n}(\frac12+ix)\in\mathbb R$ whenever $x\in\mathbb R$?

I just noticed that $B_{2n}(\frac12+ix)\in\mathbb R$, where: $x\in\mathbb R$, $n\in\mathbb N$, and $B_n(x)$ is the $n$th Bernoulli Polynomial. Why? Is there a simple, slick proof? Does it follow from ...
WillG's user avatar
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2 votes
0 answers
71 views

What is the definition of the Complete Gamma Function?

this topic is kind of confusing me a lot. I am working with a text book that defined the Complete Gamma Function as follows: $$\ln\Gamma(x) = \ln\sqrt 2\pi-x+(x-\frac{1}{2})\ln x +\frac{B_1}{1\cdot2}\...
Siebolic's user avatar
3 votes
1 answer
151 views

Closed form of $\sum_{k=1}^\infty\frac{k^nB_k}{k!}$

I developed the following: Consider $$\frac{t}{e^t-1}=\sum_{k=0}^\infty\frac{B_k}{k!}t^k,$$ where $B_k$ are the Bernoulli numbers, then $$\frac{e^t}{e^{e^t}-1}=\sum_{k=0}^\infty\frac{B_k}{k!}e^{tk}=\...
pshmath0's user avatar
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2 votes
0 answers
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Asymptotically similar functions with opposite parity, were they considered, are they useful? Case of polynomials

So, can we transform an even function into an odd function and vice versa? Let's consider this method: Transformation even->odd: Suppose $f_{even}(x)$ is a function which satisfies the following ...
Anixx's user avatar
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0 votes
2 answers
92 views

Use Bernoulli trials to select random k-length permutations

Using only Bernoulli trials with $p=0.5$, how can I efficiently create a permutation of length $k$ from a set of length $n$? I can think of a few ways to do this that waste trials, but I’d like to do ...
mrplants's user avatar
  • 105
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0 answers
77 views

Finding the expectation of a stopping time

Let $\left(X_{i}\right)_{i}$ be i.i.d. bernoulli random variables. More precisely $\mathbb{P}(X=0)=\mathbb{P}(X=1)=$ 1/2. Let $$ T=\inf \left\{n \geq 4: X_{n-3}=0, X_{n-2}=1, X_{n-1}=0, X_{n}=1\right\}...
codelearner's user avatar
0 votes
1 answer
45 views

How is possible that non independent events have the same probability?

There's a question in the book "An Introduction to Mathematical Statistics and Its Applications" that I can't understand: An urn contains $r$ red balls and w white balls. A sample of $n$ ...
maenju's user avatar
  • 343
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0 answers
408 views

The geometric distribution - How many trials occur before we obtain a success doubt

I have doubts on the same question First, What is k here? Second, I know E[X]= E[I{A}], I define indicator random variable. We ...
Encipher's user avatar
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