Questions tagged [bernoulli-polynomials]

Questions on Bernoulli polynomials and their series expansions.

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

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}{...
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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 ...
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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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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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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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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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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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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 $... 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 ... 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 , ... 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 $$E_{2n} = 2^{2n+... 1answer 231 views Series involving Bernoulli Numbers I would be interested in the following sum:$$\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 ... 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 <... 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 ... 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 ... 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} ... 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 ... 1answer 82 views Is there any answer for this Bernoulli difference? Is there any answer for this Bernoulli difference equationB_{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}^... 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$Answer: First, ... 2answers 651 views The sum of fractional powers$\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$has an extension through$...
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):=&...