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Questions tagged [bernoulli-polynomials]

Questions on Bernoulli polynomials and their series expansions.

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Sum with Bernoulli polynomial

I'm trying to prove the following identity: $$\sum_{k=0}^n \dfrac {\binom n k B_k(x)} {(n-k+1)} = x^n$$ I transformed this identity as follow: $$\dfrac{1}{(n+1)}\sum_{k=0}^n \binom {n+1} k B_k(x) = x^...
0
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1answer
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 ...
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3answers
31 views

Bernoulli Substitution for differential equation

$$x'+2t^2x=2t^2x^3 $$ I made the Bernoulli Substitution $$u=\frac{1}{x^2}$$ therefore $$u'=-2x^{-3}x'$$ then after some conversions I had the following equation $$u=4t^2u-4t^2$$ however I had the ...
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0answers
15 views

Periodic Bernoulli polynomials

How do I integrate $$\int _0^T x^m B_1(\{ x\} ) dx$$, where $\{x\} $ denotes the fractional part of $x $? Integrating by parts should work, but I can't get anything to simplify in a particularly nice ...
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0answers
23 views

Integrating periodic Bernoulli function

How to integrate functions involving periodic bernoulli function like $$\int_1^n\frac{-6}{x^4}P_{4}(x)dx$$ where $P_{4}(x) = B_{4}(x - \lfloor x \rfloor) \text{ is the 4th order periodic bernoulli ...
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0answers
13 views

Property of the remainder term in the Euler-Maclaurin formula for $\sum_{i=1}^n\log i$.

From https://en.wikipedia.org/wiki/Stirling's_approximation, I'm having trouble understanding $$R_{m,n}=\lim_{n\to\infty} R_{m,n}+O(n^{-m}).$$ I worked out $$R_{m,n}=\int_1^{n}\frac{P_m(x)}{mx^m}~ dx$$...
3
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2answers
159 views

Is there a polynomial (or series) expression for summing $S_d(a,N)=\sum_{k=0}^{N-1} \log(1+{1\over a+k \cdot d})$? (perhaps Bernoulli-type)

I need a quickly evaluatable expression for sums of consecutive logarithms of the type $$ S_{d}(a,N) = \log(1+ {1\over a})+\log(1+ {1\over a+d})+\log(1+ {1\over a+2d})+ \cdots + \log(1+ {1\over a+(N-1)...
1
vote
1answer
33 views

Property of the Bernoulli Polynomials

I'm trying to prove the following equality involving the Bernoulli polynomials: $B_{k}(x)=N^{k-1}\sum_{a=0}^{N-1}B_{k}\left(\frac{x+a}{N}\right)$ for all $N\in\mathbb{N}$. Since this is to be ...
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25 views

About Bernoulli polynomials

My question is about Bernoulli numbers and Bernoulli polynomials in the $p$-adic context. In general in fact Bernoulli numbers are defined as global object so they do not depend on $p$. If $B_k(x)$ is ...
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1answer
30 views

When solving differential equations using substitution, does it really matter which substitution you choose?

My test says to make the "appropriate substitution," but one of the differential equations can be solved as a homogenous substitution and a bernoulli's substitution. Both yield different results. Is ...
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0answers
18 views

$\sum_{i=0}^{S/2} (S/2+3/2)^{2i}\frac{2\cdot\Gamma(a) \Gamma(i+0.5)}{\sqrt(\pi) \Gamma(a+i+1)} {S \choose 2i} B_{\frac{S-2i}{2}}^{1+S}(0.5+S/2) $

Show that the following sum \begin{equation} K=\sum\limits_{i=0}^{S/2} (\frac{S}{2}+3/2)^{2i}\frac{2\cdot\Gamma(\alpha) \Gamma(i+0.5)}{\sqrt(\pi) \Gamma(\alpha + i +1)} {S \choose 2i} B_{\frac{S-2i}{...
3
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3answers
70 views

Use properties of the Bernoulli Polynomials to prove $\int_0^1 B_n(x)dx= 0$

Use the properties of the Bernoulli Polynomials to prove: $$\int_0^1 P_n(x)dx= 0 \tag{for n>0}$$ I have these properties to work with below: $$P_0(x) = 1; \qquad P'_n(x) = n P_{n-1} (x) \qquad ...
2
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1answer
69 views

Sum with a fractional number of terms???

I was playing around on desmos with the Bernoulli polynomials which are defined as $$B(x,m)=\sum_{n=0}^{m}\frac1{n+1}\sum_{k=0}^{n}(-1)^k{n\choose k}(x+k)^m$$ And I noticed that graphs were being ...
3
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0answers
96 views

Limit of an alternating series

Given some fixed integer $n$ (but one may take $n = 0$ for simplicity), I would like to compute the following limit: $$\lim_{N \to \infty}\sum_{\substack{|u_1|, |u_2|, |u_3|, |u_4| \leq N\\|u_1 + u_2 ...
4
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1answer
91 views

Integral identity involving Bernoulli polynomials

I found the following identity on Wikipedia, and I am having a difficult time proving it. For $m,n\in\Bbb N$, $$I(m,n):=\int_0^1B_n(x)B_m(x)\mathrm{d}x=(-1)^{n-1}\frac{m!n!}{(m+n)!}b_{n+m}$$ Where $...
2
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1answer
89 views

Is the series $‎\sum_{n=0}^{‎\infty‎}‎\frac{B_n(z)}{2^n}‎$ convergent?

‎$B_n(z)‎‎$‎‎ ‎is the Bernoulli polynomial, ‎we know that the series $$\sum_{n=0}^{‎\infty‎}‎\frac{B_n(z)}{n!}‎$$ ‎is ‎convergent ‎for ‎every ‎‎$z\in‎\mathbb{C}‎$‎ ‎and its‎ ‎closed form ‎is equal ...
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0answers
50 views

Help with Fourier Series $\sum_{j=1}^{\infty} \frac{1}{j^{2k}}\sin{2\pi j x}$

I've found the limit of a similar expression in the literature (Abramowitz & Stegun), which uses Bernoulli polynomials: $\sum_{j=1}^{\infty} \frac{1}{j^{2k+1}}\sin{2\pi j x}=\frac{-(-1)^k(2\pi)^{...
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4answers
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Bernoulli equation when not homogenous

$$x \frac{dy}{dx} + y = -2x^6y^4$$ Divided by x $$ \frac{dy}{dx} + \frac{y}{x} = -2x^5y^4$$ $$n = 4 $$ $$z = \frac{1}{y^3} $$ $$a(x) = \frac{1}{x}$$ $$b(x) = -2x^5 $$ $$\frac{z'}{-4+1} + \...
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1answer
45 views

Second generalized Bernoulli number $B_{2,\chi}$

Let be $\chi$ (non-trivial) Dirichlet charakter of conductor $f$. Then I know that $$B_{n,\chi}=f^{n-1}\sum_{a=1}^f\chi(a)B_n\left(\frac{a}{f}\right).$$ Assume $\chi(-1)=1$ and plug $n=2,$ then $$B_{...
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1answer
112 views

Prove following statements concerning Bernoulli polynomials

Let $B_n(x)$ be the Bernoulli Polynomial. 1) Show for $n\neq 1$ is $B_n(1)=B_n(0) (=B_n)$. 2) Determine $B_1=B_1(0)$ and $B_1(1)$. I've already tried just to plug in the values in different ...
2
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1answer
121 views

Bounds for Periodic Bernoulli Polynomials via Fourier Series

I am looking at the following result here on page 121. The Fourier series of the $p$'th Periodic Bernoulli polynomial $\mathcal{P}_p(x) := \mathcal{B}_p(\{x\})$ ($p$'th Bernoulli polynomial evaluated ...
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1answer
44 views

Prove Bernoulli Task

Prove for $\ n\ge100,n\ge50,n\ge49 \ $that the following is true: $$\ 101^{n}>100^{n}+99^{n} \ $$ Prove for $\ n\ge100,n\ge50,n\ge49 \ $ I know it’s somehow connected with Bernoulli; however, ...
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1answer
62 views

Constant determinant of matrix of Bernoulli polynomials

Let $B_n(x)$ denote the Bernoulli polynomial of degree $n$. If we construct the symmetric matrix $$ \mathbf{B}_N(x) = \begin{pmatrix} B_0(x) & B_1(x) & \cdots & B_N(x) \\ B_1(x) & B_2(...
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1answer
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First Order Differential Equations Applied Question

The population P(t) of cod in a region of the North Sea (at time t) satisfies $\frac{dP}{dt}$ = $kP(1-\frac{P}{M})-aP$ where k>0, M>0 and a>0 are constants, with k representing the breeding rate, M ...
11
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1answer
618 views

Formula for a sequence defined on $K_1(x,y) := y+0$ if $x \geq y$ and $y-1$ otherwise

Define $K_1:[0,1]^2\rightarrow\mathbb{R}$ as $$K_1(x,y) := x - \frac{1}{2} - \begin{cases} \ +(x - y - \frac{1}{2}) & \text{if $x \geq y$},\\ \ -(y - x - \frac{1}{2}) & \text{otherwise} \end{...
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0answers
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Riemann zeta meromorphic cont. using Abel summation formula

In Stein&Shakarchi, Complex Analysis, chapter 6, problem 2-3 (p. 180), they hint at a method to meromorphically continue the zeta function to the entire complex plane. I can see from Abel's ...
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0answers
56 views

Convergence of a sequence which is built with Bernoulli polynomials

The generating function of the Bernoulli Polynomials is: $$\frac{t e^{xt}}{e^t-1}=\sum\limits_{n=0}^{\infty}B_n(x)\frac{t^n}{n!}.$$ Put $b_n=B_n(1)$. Consider that $B_n(1)=B_n(0)$ for $n \geq 2$ . Now ...
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1answer
169 views

How to get the explicit formula of Bernoulli polynomial using its generating function?

I have read an article about Bernoulli Polynomial. I found that Bernoulli Polynomial has explicit formula like this: $$B_n(x)=\sum_{k=0}^{\infty}\binom nk B_kx^{n-k}$$ And the article said that the ...
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1answer
1k views

Sum of powers of natural numbers

Bernoulli stated sum of series of powers as: LINK to the image source (Power Sum) I had a doubt in the given formula in the picture! What if $n < p$ i.e. $1^4 + 2^4 + 3^4$ here $n = 3$ and $p = ...
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1answer
128 views

Integral of product of two Bernoulli polynomials

I found formula of integral of product of two Bernoulli polynomial in Takashi Agoh & Karl Dilcher (http://www.sciencedirect.com/science/article/pii/S0022247X1100312X) $\int_{0}^{1}B_k(t)B_m(t)dt=(-...
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1answer
81 views

How to prove that: $2e^{-2x}(e^x -1)= \displaystyle\sum_{n=1}^\infty \frac{\mu(n)}{\sinh(nx)}$

I'm quite lost at this, I've tried to express the $csch(nx)$ as a sum like this: $$\frac{csch(nx)}{2}= \frac{1}{e^{nx}-e^{-nx}}=\frac{1}{2nx}\sum_{k=0}^{\infty} \frac{(2nx)^k B_k(1/2)}{k!}$$ Where $...
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2answers
183 views

Bernoulli experiment - A coin toss - How to mathematically notate the experiment?

The Bernoulli random variable has a probability function: $$f_X(x) = p^x\,(1-p)^{1-x}~\mathbf 1_{x\in\{0,1\}}\\ f_X(0)=1-p\\ f_X(1)=p\qquad$$ Provide an example of a Bernoulli experiment and a ...
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1answer
201 views

Bernoulli numbers definition as power series

For any complex $x$ we define the funcitons $B_n(x)$ by the equation $$ \frac{ze^{xz}}{e^z-1} = \sum_{n=0}^\infty \frac{B_n(x)}{n!} z^n , \text{ where } |z|< 2 \pi . $$ Page 264, Apostol , ...
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1answer
87 views

quadratic sum of Bernoulli numbers

Is it possible to express $$ A_n = (2n-3)! \sum_{i=0}^{2n} \frac{B_i B_{2n-i}}{i! (2n-i)!} (2^{1-i} -1) (2^{1-2n+i} -1)(2^i -1) $$ in a more compact form, perhaps using the identity $$ E_{2n} = 2^{2n+...
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1answer
231 views

Series involving Bernoulli Numbers

I would be interested in the following sum: $$\sum^n_{k=1} {{n}\choose{k}} \frac {B_{k+1}} {(k+1)!}$$ It would be great to get a closed form for it. The problem is that I do not know what to do ...
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1answer
717 views

Probabilies of rolling n dice to add up to a specific sum

I try to visualize a bernoulli chain. With variables p q and s. p probability for success <...
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2answers
60 views

A confusion in Bernoulli equation solution

I have a bernoulli equation with its proper and correct solution written step by step. The issue is that it contains a line that is transformed to the 2nd line which I was unable to understand how it ...
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0answers
375 views

Euler Maclaurin formula proof

I'm trying to derive the Euler Maclaurin formula following steps provided in Bender and Orszag's Advanced Mathematical Methods for Scientists and Engineers problem 6.88 page 315. Given a sum of the ...
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1answer
91 views

Misunderstanding about Bernoulli-Euler relation?

It is well - known (and famous) that the identity \begin{align} E_{n - 1} \left( x \right) = \frac{2}{n}\left[ {B_n \left( x \right) - 2^n B_n \left( {\frac{1}{2}x} \right)} \right] \\ \end{align} ...
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1answer
150 views

Stirling numbers and bernoulli numbers for summing up n numbers to the kth power

I am currently working on problem 487 on project euler. I did some research and I only see 2 possibilities to solve this problem: 1. By using faulhabers formula 2. by using the formula featuring the ...
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1answer
82 views

Is there any answer for this Bernoulli difference?

Is there any answer for this Bernoulli difference equation $$B_{n+1}(x)-B_n(x)=?$$ where $B_n(x)$ is the Bernoulli polynomial defined by the exponential generating function$${ze^{xz} \over \mathrm{e}^...
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0answers
48 views

Bernoulli Differential Equations

I'm working through some Bernoulli differential equation exercises and wanted to check my work on one and get some guidance on one that is totally stumping me. 1.) $u' + u = e^xu^4$ Answer: First, ...
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2answers
651 views

The sum of fractional powers $\sum\limits_{k=1}^x k^t$.

This post is a continuation of Generalization of the Bernoulli polynomials ( in relation to the Index ) , the definition of the Bernoulli polynomial $B_t(x)$ with $|x|<1$ has an extension through $...
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1answer
153 views

Generalization of the Bernoulli polynomials ( in relation to the Index )

The Riemann $\zeta$-function is here with analytical extension (e.g. with her functional equation). Definition for $|x|<1$ and $-t\in\mathbb{R}\setminus\mathbb{N}$: \begin{align*} B_t(x+1):=&...