Questions on Bernoulli polynomials and their series expansions.
Questions on Bernoulli polynomials and their series expansions, also in combination with the Riemann zeta function.
Questions tagged [bernoulli-polynomials]
84 questions
3
votes
2 answers
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 "expanded out as a formal power series"
Since we…
gist076923
- 1,266
3
votes
1 answer
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}
holds for all $n \ge 1$ and $0 \le x \le 1$.…
Mohammad W. Alomari
- 1,165
1
vote
0 answers
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 way so I can't work out how to make an induction…
tomos
- 1,650
0
votes
1 answer
Can we write $B_n(x)-B_n(0)=nx^{n-1}$?
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 change the subscripts to be $B_{n+1}$ and $B_n$ instead…
MAS
- 10,638
0
votes
0 answers
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 $P_{4}(x) = B_{4}(x - \lfloor x \rfloor) \text{ is the 4th order periodic bernoulli function}$ ?
Here's my attempt, please verify if…
jnxd
- 97
- 1
- 10
0
votes
1 answer
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 kinds of Bernoulli Polynomial representations (with…
mojo
- 13
-1
votes
1 answer
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, I'm having trouble figuring out how to do this.
Valerie
- 33