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.
368
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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 ...
- 7,426
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2
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50
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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 ...
- 105
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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
$$
T=\inf \left\{n \geq 4: X_{n-3}=0, X_{n-2}=1, X_{n-1}=0, X_{n}=1\right\}...
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1
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How is possible that non independent events have the same probability?
There's a question in the book "An Introduction to Mathematical Statistics and Its Applications" that I can't understand:
An urn contains $r$ red balls and w white balls. A sample of $n$ ...
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The geometric distribution - How many trials occur before we obtain a success doubt
I have doubts on the same question
First, What is k here?
Second, I know E[X]= E[I{A}], I define indicator random variable. We ...
- 179
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Proof of sum with Bernoulli poly: $\frac{(-1)^{n+1}}{2}\sum_{k=0}^{n-1} \binom{n-1}{j} B_{n+j}(1/2) \frac{1}{n+j} =4^{-n}-\frac{1}{2n\binom{2n}{n}}$
The following sum is implied by some work I performed with infinite series.
$$ \frac{(-1)^{n+1}}{2}\sum_{k=0}^{n-1} \binom{n-1}{j} B_{n+j}(1/2) \frac{1}{n+j} =4^{-n}-\frac{1}{2n\binom{2n}{n}}$$
The $...
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Conjectured analogue of Fermat's Little Theorem for Bernouli numbers
Is the following analogue of Fermat's Little Theorem for Bernouli numbers true?
Let $D_{2n}$ be the denominator of $\frac{B_{2n}}{4n}$ where $B_n$ is
the $n$-th Bernoulli number. If $\gcd(a, D_{2n}) =...
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Power series of $x/(1-ae^{-x})$.
I am looking for a power series expansion for the function $x(1-ae^{-x})^{-1}$ (perhaps for $0<a<1$).
Using the Bernoulli numbers, I can write
\begin{align*}
\frac{x}{1-ae^{-x}} &= \frac{x}{...
- 329
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1
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Formula for Stirling numbers expressed with Bernoulli numbers?
Is there a formula which expresses any of the Stirling numbers (1st or 2nd kind) in terms of the Bernoulli numbers? For example, here is the reverse
$$
B_k=\sum_{m=0}^{k} (-1)^m \frac{m!}{m+1}\sigma(k,...
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Efficiency of two different estimators (Bernoulli)
I am using a book, where the following exercise appears:
Two estimators for a random sample of size $n$ (Bernoulli population):
$T_1=\frac{\sum\limits_{i=1}^n X_i + 2X_n}{n+2}$
$T_2=\frac{\sum\limits_{...
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An equation by the definition of Bernoulli number
I am working on Bernoulli number. I learnt the definition of Bernoulli number on the book by a Japanese mathematician. The name of the book is Number Theory 1: Fermat's dream. The book defines the ...
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Functions with Bernoulli Number Product Coefficients
I am trying to construct a function with 2 Bernoulli Numbers Product.
For example we have a following for cotangent:
$$\cot(z)=\sum _{k=1}^{\infty } \frac{(-1)^k 2^{2 k} B_{2 k} z^{2 k-1}}{(2 k)!}+\...
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Normal approximation to Bernoulli variable
I'm looking for a normal approximation for a Bernoulli variable (so I can later sum multiple correlated approximated variables)
The trivial approximation is taking the mean and variance of the ...
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About the functional equation between $\zeta$ and $\Gamma$?
It is well-known applying the monotone convergence theorem that for all $x>1$ we have the following functional equation : $\zeta(x)\Gamma(x)= \displaystyle \int \limits_{[0,+\infty]} \dfrac{t^{x-1}}...
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Show that $f(x)<1$ for a special $x$
Let define the function :
$$f\left(x\right)=\frac{2}{x\left(\tanh\left(xe^{-1}\right)+1\right)}$$
Show that :
$$f\left(\frac{1+\sqrt{3}}{2}\right)<1$$
Some facts :
$$\tanh(x)=\sum_{n=1}^{\infty}\...
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Finding the conditional mean of a Bernoulli and Standard Normal Random Variable in the same equation
Equation / Necessary Information
Consider the following Model $$Y_i = X_{1i} + u_i$$
where $X_{1i}$ is the Bernoulli randiom variables with equal probabilities and $u_{i}$ is the standard normal ...
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Bernoulli Polynomials and Euler summation
I'm given the Euler summation formula in it's simplest terms, namely
$$
\sum_{v = 0}^{n} f(v)=\frac{1}{2}\left( f(n)+f(0) \right)
+ \int_{0}^{n} f(x)\textrm{d}x + \int_{0}^{n} \left(
x - \...
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1
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A polynomial rep. for $\frac{1}{m!} \sum_{k=0}^n (-1)^{k+m} \frac{k^n}{k+1} \binom{m}{k}$
Let the generalized Bernoulli polynomials be defined by their generating function
$$ \sum_{n=0}^\infty \frac{B_n^{(s)}(x)}{n!} t^n = \Big(\frac{t}{e^t-1}\Big)^s\ e^{xt} $$
The Stirling numbers of the ...
- 6,493
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Sum a product of Bernoulli numbers and binomial coefficients
Context: I am interested in developing the large-$x$ asymptotic series of the digamma function
$$\psi\Big(\frac{1}{2}+ix\Big)$$
for real positive $x$.
For this I am using the known asymptotic ...
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answer
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Is the determinant of a Hessenberg matrix whose elements are the Bernoulli numbers positive?
The Bernoulli numbers $B_r$ are generated by
\begin{equation}\label{Bernoullu=No-dfn-Eq}
\frac{z}{e^z-1}=\sum_{r=0}^\infty B_r\frac{z^r}{r!}
=1-\frac{z}2+\sum_{r=1}^\infty B_{2r}\frac{z^{2r}}{(2r)!}, \...
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Evaluating $\int_0^y \frac{z^2e^{az}}{e^z-1}\,\mathrm dz$.
Given $a,y\in\mathbb{C}$, I want to find a closed form for
$$
\int_0^y \frac{z^2e^{az}}{e^z-1}\,\mathrm dz.
$$
Background:
This integral was obtained from an infinite sum involving Bernoulli number ...
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1
answer
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Two properties of the set of Bernoulli numbers
Let $\mathcal{B}$ be the set of all Bernoulli numbers. We are looking for an answer for
one of the following (or both) questions.
(a) Is there an infinite subset $S$ of rationals such that
$(\mathcal{...
- 2,153
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Residue of $\dfrac{\mathrm{e}^\frac{2}{z}}{\sin z}$ at $0$
Is there a nice way to find (and represent) the residue of the function $\dfrac{\mathrm{e}^\frac{2}{z}}{\sin z}$ at $0$?
I tried using the product of the associated Laurent series for $\mathrm{e}^\...
- 1,771
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0
answers
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Dirichlet series of Bernoulli polynomials
I am interested in the Dirichlet series
$$
\mathcal{B}(k,s) = \sum_{m\geq 1} \frac{B_k(m)}{m^s}
$$
where $B_k(x)$ is the $k$th Bernoulli polynomial and $\Re(s) > k + 1$. This converges for all $k$ ...
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What's the nicest proof of the formula for the sum of the $k$-th powers of the first natural numbers?
Do you know of a text where I can find a nicely motivated proof of the formula for $1^{k}+2^{k}+\cdots+n^{k}$?
At the very beginning of page 68 of Professor H. S. Wilf's generatingfunctionology, one ...
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Merged Poisson Process Rate
If I have two process $X_t$ with rate λ and $Y_t$ with rate μ and I merge them I get an overall rate of λ + μ.
Is this the same logic that would follow if I have a process of $2X_t + 3Y_t$ to get 2λ + ...
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Correlation between union of correlated Bernoulli processes
Let $X_{1,1},X_{1,2},X_{2,1},X_{2,2}$ be identically distributed r.v.s with distribution $\sim Be(p)$ and equally par-wise correlated, with pair-wise Pearson correlation coefficient $\rho$, e.g. $Corr[...
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Conjecture on bernoulli numbers and binomial coefficients
In playing around with some formulas, I have come up with the following conjecture. I have checked it for a lot of cases, and have good reason to believe it to be true. If anyone could help, I would ...
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Getting $B_4$ from the recursive definition $B_n=1-\sum_{k=0}^{n-1}{n\choose k}\frac{B_k}{n-k+1}$
Background
I've been learning the basics of Bernoulli numbers and this is a reference to What is the simplest way to get Bernoulli numbers? and vadim123's succinct reply, where he stated:
The ...
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Finding an integral using Bernoulli Numbers exponential generating function
The following exponential generating function of the Bernoulli Numbers is the following:
$$\frac{t}{e^t-1}=\sum_{n=0}^{\infty} B_n \frac{t^n}{n!}$$
The following result, which is a well known special ...
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What is the smallest number of tosses for all outcomes?
What is the smallest number of tosses that need to be done to get all the possible outcomes $\{1, 2, 3, 4, 5,6\}$ of a right dice with a reliability of $0.99$?
My attempt is to use the Bernoulli ...
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1
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Closed form for sums containing exponential function: [closed]
How to get closed form of following sums:
$$\sum_n e^{-n/2}n^{k-1}\left(s-\frac{1}{mn}\right)^k$$
$$\sum_n \frac{e^{-n/2}}{n^2\left(s-\frac{1}{mn}\right)}$$
Here $s,k,m$ are constants and $n$ runs ...
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Proving Property of Bernoulli's polynomial $B_n(x+1)-B_n(x)=nx^{n-1}$
I am trying to show that $B_n(x+1)-B_n(x)=nx^{n-1}$, where $B_n(x)$ is the Bernoulli polynomial. In order to avoid circular reasoning, I define the Bernoulli's number $B_n$ as the coefficient in the ...
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Relationship between Bernoulli numbers of first and second type
I am interested in an expresion that allows to calculate Bernoulli numbers of the first kind $B_n$ from the values of the Bernoulli numbers of the second kind $b_n$ (also known as Cauchy numbers of ...
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Convergence of Bernoulli numbers infinite sum
In https://en.wikipedia.org/wiki/Harmonic_number I found that harmonic numbers admit asymptotic expansion as:
$$H_n \approx \ln n + \gamma_0 + \frac{1}{2n} - \sum_{k=1}^{\infty}{\frac{B_{2k}}{2kn^{2k}}...
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How to compute the gradient inside the Expectation
Assume, we have a function
$$
\mathcal{E}(\phi)=\mathbb{E}_{z \sim \text { Bernoulli}(\sigma(\phi))}[f(z)]
$$
Here $z$ is a variable, and follow the Bernoulli distribution with parameters $\sigma(\phi)...
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2
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Forward difference of Bernoulli polynomials
I am working on a research project on Bernoulli polynomials, which are defined as
$$B_n(x) = \sum_{j} \binom{n}{j}B_j x^{n-j}$$
where $B_j$ are the Bernoulli numbers. There is also the "...
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How to prove that $\sum_{i=1}^{m} 2^{i} B_{i}\binom{m}{i} \frac{1}{m-i+1}=\frac{2 m+1}{m+1}$ when $m$ is an even positive integer?
I want to prove the identity below $$
\sum_{i=1}^{m} 2^{i} B_{i}\binom{m}{i} \frac{1}{m-i+1}=\frac{2 m+1}{m+1}
$$
I've carried out some computations and verified it but it only seems to hold when $m$ ...
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How is $ B_n = 1- \sum_{k=0}^{n-1} \binom{n}{k} \frac{B_k}{n-k+1} $ Where $B_n$ are the Bernoulli Numbers with $B_1 = \frac{1}{2}$
So I was browing Wikipedia just looking at Bernoulli Number identities and I stumbled across this
$$ B_m^+ = 1- \sum_{k=0}^{m-1} \binom{m}{k} \frac{B_k^+}{m-k+1} $$
The Wikipedia page said that this ...
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Can the Bernoulli numbers be viewed as a 'renormalization' of a finite geometric series with term $e^{-x}$, by integrating over $(-1,0)$?
I was playing around with ways to calculate Bernoulli numbers; for this post I will take their generating function as $x/(1-e^{-x})$, that is,
$$
\sum_{n=0}^{\infty} \frac{B_n}{n!}x^n = \frac{x}{1-e^{-...
- 8,136
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2
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133
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Proving that Bernoulli Number $B_0 =1$
This question was given by our instructor today to do by ourselves and I am having trouble proving it. So, I am asking here.
Definition of Bernoulli Polynomials : For any complex number x we define ...
- 1,348
1
vote
1
answer
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Expansion of the hyperbolic cotangent
I have this question:
Prove that $$\coth(z) = \sum_{n=0}^\infty {B_{2n} \over 2(2n)!}(2z)^{2n−1}$$
$\forall |z| < π$.
I have already obtained the expression, but I don't know how to get the |z| <...
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0
answers
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Complex Analysis and Bernoulli Numbers from $\frac{z}{2} \cot (\frac{z}{2})$
Define the Bernoulli numbers $B_n$ by $\frac{z}{2} \cot (z/2) = 1 - B_1 \frac{z^2}{2!} - B_2 \frac{z^4}{4!} - B_{3} \frac{z^6}{6!} - ...$ Explain why there are no odd terms in this series. What is the ...
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extension of $\zeta(s)=\lim_{n\to\infty}\sum_{k=1}^n 1/k^s -\int_0^n 1/x^s$
I saw $\zeta(1/2)=\lim_{n\to\infty}\sum_{k=1}^n 1/\sqrt k-\int_0^n\ 1/\sqrt x\ dx$
here
and similarly
$\zeta(s)=\lim_{n\to\infty}\sum_{k=1}^n 1/k^s -\int_0^n 1/x^s$ for $0<s<1$.
Rewriting this ...
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1
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364
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Non-trivial zero(s) of Akiyama-Tanigawa triangle
Introduced in 1997, the Akiyama-Tanigawa triangle is a doubly-indexed recursion that encodes the Bernoulli numbers, among other sequences. It is defined as follows: let $a:\mathbb{N^0}\times\mathbb{N^+...
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1
answer
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Different initial condition in a recursive formula for Bernoulli numbers
Consider the following recursive equation for $k\ge 1$:
\begin{equation}
a_k = \frac{1}{2} + \sum_{i=0}^{k-1} \frac{-1}{2k + 1 - 2i} \binom{2k}{2i} a_i \tag{1}.
\end{equation}
If $a_0 = 1$, then $a_k$...
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1
answer
29
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Bernoulli random variables choice of values
It seems that typically the standard is to define a bernoulli random variable $X$ as
$$
X = \begin{cases}
1 & \text{with probability } p \\
0 & \text{with probability } 1 - p
\end{cases}
$$
My ...
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4
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Is it obvious intuitively that $1^p + 2^p + \cdots (n-1)^p$ is a polynomial in $n$?
I am reading about Bernoulli function in "Calculus vol.1" by Matsusaburo Fujiwara(in Japanese).
The author proved that $$1^p + 2^p + \cdots (n-1)^p$$ is a polynomial in $n$ of degree $p+1$....
- 6,093
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0
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78
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Another sum involving Benoulli numbers
In a recent question that I am still working on, I needed to consider some sums involving Bernoulli numbers. After a lot of computations and simplifications, I ended up having to manipulate the ...
- 1,137
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0
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Faulhaber's polynomials and irreducibility in the sums of powers
$\color{brown}{\textbf{The setup and observations}}$
For $d\in{\bf N}$ and $n\in{\bf N}^\star$, denote by $S_d(n)$ the sum of the $d$-powers of integers $1$ to $n$: $$S_d(n)=\sum_{k=1}^nk^d.$$
A ...
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