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.

Filter by
Sorted by
Tagged with
2 votes
0 answers
18 views

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 ...
user avatar
  • 7,426
0 votes
2 answers
50 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 ...
user avatar
  • 105
0 votes
0 answers
44 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\}...
user avatar
0 votes
1 answer
42 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 $n$ ...
user avatar
  • 145
0 votes
0 answers
41 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 ...
user avatar
  • 179
4 votes
0 answers
230 views

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 $...
user avatar
  • 6,493
0 votes
2 answers
173 views

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}) =...
user avatar
1 vote
1 answer
64 views

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}{...
user avatar
0 votes
1 answer
120 views

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,...
user avatar
0 votes
0 answers
31 views

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_{...
user avatar
  • 101
2 votes
1 answer
92 views

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 ...
user avatar
0 votes
0 answers
33 views

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)!}+\...
user avatar
0 votes
0 answers
28 views

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 ...
user avatar
0 votes
1 answer
85 views

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}}...
user avatar
  • 2,979
0 votes
1 answer
93 views

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}\...
user avatar
0 votes
0 answers
19 views

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 ...
user avatar
1 vote
0 answers
56 views

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 - \...
user avatar
  • 435
1 vote
1 answer
50 views

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 ...
user avatar
  • 6,493
5 votes
2 answers
116 views

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 ...
user avatar
2 votes
1 answer
78 views

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)!}, \...
user avatar
  • 997
2 votes
1 answer
152 views

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 ...
user avatar
  • 700
4 votes
1 answer
92 views

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{...
user avatar
1 vote
0 answers
52 views

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}^\...
user avatar
  • 1,771
1 vote
0 answers
71 views

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 $k$ ...
user avatar
  • 73
1 vote
1 answer
88 views

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 ...
user avatar
  • 527
0 votes
1 answer
28 views

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 + ...
user avatar
0 votes
1 answer
76 views

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[...
user avatar
  • 9
1 vote
0 answers
167 views

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 ...
user avatar
0 votes
1 answer
89 views

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 ...
user avatar
  • 1,482
5 votes
0 answers
75 views

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 ...
user avatar
1 vote
2 answers
34 views

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 ...
user avatar
  • 1,239
-1 votes
1 answer
85 views

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 ...
user avatar
  • 15
0 votes
1 answer
84 views

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 ...
user avatar
0 votes
0 answers
25 views

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 ...
user avatar
1 vote
1 answer
95 views

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}}...
user avatar
0 votes
0 answers
49 views

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)...
user avatar
  • 753
1 vote
2 answers
180 views

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 "...
user avatar
  • 76
0 votes
2 answers
67 views

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 $m$ ...
user avatar
0 votes
2 answers
96 views

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 ...
user avatar
3 votes
0 answers
78 views

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^{-...
user avatar
0 votes
2 answers
133 views

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 ...
user avatar
  • 1,348
1 vote
1 answer
104 views

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| <...
user avatar
  • 23
0 votes
0 answers
155 views

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 ...
user avatar
  • 7,072
1 vote
1 answer
73 views

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 ...
user avatar
  • 757
14 votes
1 answer
364 views

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^+...
user avatar
1 vote
1 answer
63 views

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 $a_k$...
user avatar
  • 654
1 vote
1 answer
29 views

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 ...
user avatar
-1 votes
4 answers
112 views

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 $p+1$....
user avatar
  • 6,093
0 votes
0 answers
78 views

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 ...
user avatar
7 votes
0 answers
303 views

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 ...
user avatar

1
2 3 4 5
8