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Questions tagged [bernoulli-polynomials]

Questions on Bernoulli polynomials and their series expansions.

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Connection between the polylogarithm and the Bernoulli polynomials.

I have been studying the polylogarithm function and came across its relation with Bernoulli polynomials, as Wikipedia site asserts: For positive integer polylogarithm orders $s$, the Hurwitz zeta ...
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Proving that $\lim_{ n \to \infty} \left\{ \frac{1}{2^{m n}} \sum_{r=1}^{2^n-1}(-1)^r r^m\right\}=-\frac{1}{2}$ independently of the value of $m$.

It seems that, independently of the value of $m$, we have $$ \lim_{ n \to \infty} \left\{ \frac{1}{2^{m n}} \sum_{r=1}^{2^n-1}(-1)^r r^m\right\}=-\frac{1}{2} $$ I've tested it numerically but I have ...
Neves's user avatar
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How to derive this polylogarithm identity (involving Bernoulli polynomials)?

How can one derive the following identity, found here, relating the polylogarithm functions to Bernoulli polynomials? $$\operatorname{Li}_n(z)+(-1)^n\operatorname{Li}_n(1/z)=-\frac{(2\pi i)^n}{n!}B_n\!...
WillG's user avatar
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Integral expression involving Bernoulli polynomial that allows to sum a trigonometric integral series.

Find a integral expression involving even degree Bernoulli polynomials that allows sum the series: $$P(k)=\sum_{n=1}^{\infty}\dfrac{\operatorname{Si}(n\pi)}{n^{2k+1}}$$ where Si(x) denotes the sine ...
Sergio Ferrer's user avatar
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Approximation of the Bernoulli periodic function

I remember seeing a paper that provided a summation approximation of the Bernoulli periodic function which converges when $p\ge 2$; $$\dfrac{P_{p}(x)}{(p!)}$$ but I don’t quite remember it, I know for ...
Smithy's user avatar
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The operator $\frac{D}{e^D-1}$ in definition of Bernoulli polynomials

I am trying to understand Bernoulli polynomials, and so I came across this abomination in the article: $$ B_n(x) = \frac{D}{e^D-1} x^n $$ where $D$ is the differentiation operator and the fraction is &...
gist076923's user avatar
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$n$-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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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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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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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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Approximation of Dirichlet Series over Bernoulli Polynomials

In this post, the questioner asked about the behavior of a Dirichlet series over Bernoulli polynomials: $$ \mathcal{B}(k,s) = \sum_{m\geq 1} \frac{B_k(m)}{m^s}. $$ They show that this sum is equal to $...
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Closed formula for variation of Fourier series of Bernoulli polynomials

The Fourier series for the periodic Bernoulli polynomials $$ \sum_{k \in \mathbb{Z}-\{0\}} \frac{e^{2\pi ikx}}{k^n} = - \frac{(2\pi i)^n}{n!} P_n(x), \hspace{0.5cm} n \geq 1 $$ is well known. I am ...
Dave's user avatar
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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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FUN with f̶̶l̶̶a̶̶g̶̶s̶ Newton Cotes Quadrature formula and Bernoulli polynomials of the second kind

I was told to phrase my question in a more exciting way when I asked it last time. The following is a preliminary consideration. If you don't need it, just scroll down to START HERE. Here we go then: ...
Physics's user avatar
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Fourier coefficients of Bernoulli periodic polynomials in terms of sine and cosine.

I want to express the Fourier Coefficients of the Bernoulli Polynomials in terms of sine or cosine, where $$B_n: S^1 \rightarrow \mathbb{R}$$ with $S^1$ identified to $[0,1)$ (hence polynomials of ...
Ianatore's user avatar
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4 votes
1 answer
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Solution containing Riemann Zeta function for an integral involving the EGF of the Bernoulli/Euler polynomials

In this post, the first of the following integrals is questioned. I added the second one. $$ \begin{align} &2\int_{0}^{\infty}\left(\sum_{k=0}^{n}\frac{\left(-1\right)^{k}B_{k}(1)}{k!}x^{k-n-1}-\...
tyobrien's user avatar
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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 - \...
Sebastian's user avatar
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Sum with Bernoulli poly. $\sum_{k=0}^{\lfloor n/2 \rfloor}\binom{n+1}{2k} B_{2k}(1/4) $

I have a roundabout proof for $$ \sum_{k=0}^{\lfloor n/2 \rfloor}\binom{n+1}{2k} B_{2k}(1/2)= \frac{n+1}{2^n} $$ where $B_{2k}(x)$ are the Bernoulli polynomials,$B_0(x)=1, B_2(x)=x^2-x+1/6,$ etc. ...
user321120's user avatar
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2 votes
2 answers
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Elementary method to prove that $|\overline B_{2n}(x)|\le|B_{2n}|$?

I am currently trying to estimate the error term in Euler-Maclaurin summation formula and need to establish an upper bound for the periodic Bernoulli polynomial $\overline B_{2n}(x)=B_{2n}(x-\lfloor x\...
TravorLZH's user avatar
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Recurrence with Bernoulli-Barnes Polynomials

The Bernoulli-Barnes polynomials $B_n^{(2)}(x;a_1,a_2)$ with generating function $$ \sum_{n\geqslant0}\frac{B_n^{(2)} (x;a_1,a_2)}{n!} z^n = \left(\frac{a_1 z}{e^{a_1 z}-1}\right)\left(\frac{a_2 z}{e^{...
Permutator's user avatar
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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 $k$ ...
mfox's user avatar
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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 ...
Jamai-Con's user avatar
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2 answers
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The alternating Fourier series associated with the fourth Bernoulli polynomial

The Fourier series $\sum_{n\in\mathbb Z\setminus \{0\}} \frac{\cos(2\pi n t)}{n^4}$ converges on $[0,1]$ to $-\frac{2^4}{4!}\pi^4B_4(t)$, where $B_4(t)=t^4-2t^3+t^2-1/30$ is the fourth Bernoulli ...
ray's user avatar
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Is it a well-known sequence of polynomials?

Let consider $(P_n)_{n\ge 0}$ a sequence of polynomials given by : $\forall t \in \mathbb{R}$; $\ P_0(t)=1$. $\forall n\ge 1$; $\ (P_n)'=P_{n-1}$. $\forall n\ge 1$; $\displaystyle \int_{0}^{1} P_n(t) \...
Maman's user avatar
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1 answer
100 views

Is it possible to prove that $\sum_{k=1}^{\infty}\frac{\sin(kx)}{k^a}\geq 0$ for $a\geq 1$ and $x\in [0,\pi]$?

I was trying to prove that $$\sum_{k=1}^{\infty}\frac{\sin(kx)}{k^a}\geq 0,$$ for $0\leq x\leq \pi$, and $a\geq1$. For $a$ being an odd integer, this is not really a problem, as the sums may then be ...
R.J.  Etienne's user avatar
1 vote
1 answer
202 views

Formula for alternating sum of odd powers of consecutive integers

I am trying to work through this identity on page 151 of Paulo Ribenboim's "13 Lectures on Fermat's Last Theorem" $\sum_{j=1}^n (-1)^{j-1} j^{2k+1} = (-1)^{n+1}\{ \frac{n^{2k+1}}{2} + \binom{...
L3582's user avatar
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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 ...
Howardli621's user avatar
2 votes
3 answers
301 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 "...
Joe Bob's user avatar
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0 votes
2 answers
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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 $m$ ...
Instagram-creative_math_'s user avatar
1 vote
2 answers
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Question in Ch-12 Apostol's Number theory (Vol1)

I am trying some exercises from Apostol's Introduction to Analytic number theory and I could not solve this particular problem (number 16) of textbook and need help. I am sorry, I wouldn't be able to ...
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-1 votes
4 answers
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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 $p+1$....
tchappy ha's user avatar
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7 votes
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Sum $\sum_{(k_1, k_2, k_3): k_1+k_2+k_3=K, \,\, n_1+n_2+n_3=N}k_1^{n_1}\times k_2^{n_2} \times k_3^{n_3}$

Let $k_1, k_2, k_3$ be natural non-negative numbers such that $k_1+k_2+k_3=K$. Let $n_1, n_2, n_3 \in \{0, \ldots, N\}$ and such that $n_1+n_2+n_3=N$. Calculate $$ S=\sum_{(k_1, k_2, k_3): k_1+k_2+k_3=...
user4164's user avatar
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1 answer
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Are the Bernoulli numbers $B_{2n}$ given by an elementary function of $n$?

I found an old question asking for a proof that the factorial function is nonelementary, and the Claim 2 section of the answer there (by Vincenzo Oliva) doesn't quite make sense to me: https://math....
Davey's user avatar
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1 answer
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Verifying Big O of remainder term when deriving Stirling's approximation formula

I am trying to verify the following statement mentioned in an wikipedia article (https://en.wikipedia.org/wiki/Stirling%27s_approximation#Derivation), the statement is used to derive the stirling's ...
jnxd's user avatar
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2 votes
0 answers
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Does $\sum_{n=0}^{\infty}\frac{B_{n-1}}{n!}B_n(z)$ have closed form?

I'm trying to evaluate the closed form of $$\sum_{n=0}^{\infty}\frac{B_{n-1}}{n!}B_n(z),$$ if there exists, when the generating function of the Bernoulli Polynomials is: $$\frac{t e^{xt}}{e^t-1}=\sum\...
shabnam's user avatar
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1 answer
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Probabilities with ${ n \choose k}$

can i convert $\sum_{k=0}^n$ in to ${a \choose b}$ form in the Bernoulli Equation shown below: $Pr[k\mbox{ successes in }n\mbox{ trials }] =\sum_{k=0}^n \binom{n}{k}s^kf^{n-k}$ , $s$ and $f$ are ...
user3696623's user avatar
3 votes
1 answer
93 views

integral representation for $\sum_{k=0}^{x}k^{p}$

How the following integral representation can be derived? $$\sum_{k=0}^{x}k^{p}=\int_{0}^{x+1}B_{p}\left(t\right)dt=\frac{B_{p+1}\left(x+1\right)-B_{p+1}}{p+1}$$ I know Faulhaber's formula which is ...
user avatar
1 vote
1 answer
274 views

The Bernoulli Polynomials

We know that where $B_n(t)$ is Bernoulli polynomials. My question: Can Bernoulli polynomials be orthogonalized with respect to a weight function $\omega$? or I mean what is a weight function under ...
HD239's user avatar
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1 answer
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Can we write $B_n(x)-B_n(0)=nx^{n-1}$? [closed]

Bernouli polynomials satisfies the relation $B_n(x+1)-B_n(x)=nx^{n-1}$. Can we write $B_n(x)-B_n(0)=nx^{n-1}$ or something like $B_{n+1}(x)-B_n(0)$ to be equal to $nx^{n-1}$? I mean I want to ...
MAS's user avatar
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1 vote
1 answer
108 views

How to associate Bernouli numbers or Bernouli polynomials into the relation $ \sum_{n=0}^{\infty} \left[n(4x-1)+(2x) \right]x^n=0$?

How to associate Bernouli numbers or Bernouli polynomials into the relation \begin{equation} \sum_{n=0}^{\infty} \left[n(4x-1)+(2x) \right]x^n=0, \ \ \ \cdots \cdots (1) \end{equation} and \begin{...
MAS's user avatar
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4 votes
0 answers
117 views

How to describe $\overset{\sim}{B}_n(x):=\sum_{k=0}^n\binom{n}{k}B^-_{n-k}H_kx^k$ and in particular $\overset{\sim}{B}_n(1)$?

Denote by $B_n(x)=\sum_{k=0}^n\binom{n}{k}B^-_{n-k}x^k$ the $n$-th Bernoulli polynomial, where $B^-_0=1,B^-_1=-\frac{1}{2},B^-_2=\frac{1}{6},...$ are the Bernoulli numbers. Im interested in describing ...
Redundant Aunt's user avatar
5 votes
2 answers
230 views

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^...
Katy's user avatar
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0 votes
1 answer
128 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 ...
Katy's user avatar
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0 votes
3 answers
107 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 ...
keiro's user avatar
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1 vote
0 answers
143 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 ...
tomos's user avatar
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0 votes
0 answers
89 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 ...
jnxd's user avatar
  • 97
2 votes
0 answers
210 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 $$R_{m,n}=\int_1^{n}\frac{P_m(x)}{mx^m}~ dx$$...
Abraham Zhang's user avatar
3 votes
2 answers
221 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)...
Gottfried Helms's user avatar
1 vote
1 answer
165 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 ...
user608571's user avatar
0 votes
1 answer
61 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 ...
Melanie's user avatar
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