# 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 ...
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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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### 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 ...
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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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### 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 $m$ ...
1 vote
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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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### 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 $p+1$....
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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 ...
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### 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 ...
1 vote
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### 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 ...
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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 ...
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### 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 $$\sum_{n=0}^{\infty} \left[n(4x-1)+(2x) \right]x^n=0, \ \ \ \cdots \cdots (1)$$ and \begin{...
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### 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 ...
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### 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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