Question

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?

score 84 · Answer 1 · 2020-11-25 20:09:21

84

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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?

Published on 25 Nov 2020 - 20:09

score 290 · Answer 2 · 2020-12-11 21:53:40

290

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Forward difference of Bernoulli polynomials

Published on 11 Dec 2020 - 21:53

score 250 · Answer 3 · 2021-01-19 13:15:49

250

Views

Proving Property of Bernoulli's polynomial $B_n(x+1)-B_n(x)=nx^{n-1}$

Published on 19 Jan 2021 - 13:15

score 195 · Answer 4 · 2021-02-16 16:20:44

195

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Formula for alternating sum of odd powers of consecutive integers

Published on 16 Feb 2021 - 16:20

score 100 · Answer 5 · 2021-02-17 13:55:47

100

Views

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

Published on 17 Feb 2021 - 13:55

score 74 · Answer 6 · 2021-03-16 14:48:22

74

Views

Is it a well-known sequence of polynomials?

Published on 16 Mar 2021 - 14:48

score 143 · Answer 7 · 2021-04-17 11:25:41

143

Views

The alternating Fourier series associated with the fourth Bernoulli polynomial

Published on 17 Apr 2021 - 11:25

score 110 · Answer 8 · 2021-05-05 01:16:12

110

Views

What's the nicest proof of the formula for the sum of the $k$-th powers of the first natural numbers?

Published on 05 May 2021 - 1:16

score 129 · Answer 9 · 2021-05-12 18:09:49

129

Views

Dirichlet series of Bernoulli polynomials

Published on 12 May 2021 - 18:09

score 47 · Answer 10 · 2021-05-15 03:47:57

47

Views

Recurrence with Bernoulli-Barnes Polynomials

Published on 15 May 2021 - 3:47

score 121 · Answer 11 · 2021-05-26 15:14:51

121

Views

Elementary method to prove that $|\overline B_{2n}(x)|\le|B_{2n}|$?

Published on 26 May 2021 - 15:14

score 75 · Answer 12 · 2021-07-15 21:23:44

75

Views

Sum with Bernoulli poly. $\sum_{k=0}^{\lfloor n/2 \rfloor}\binom{n+1}{2k} B_{2k}(1/4) $

Published on 15 Jul 2021 - 21:23

score 85 · Answer 13 · 2021-08-04 19:00:37

85

Views

Bernoulli Polynomials and Euler summation

Published on 04 Aug 2021 - 19:00

score 105 · Answer 14 · 2021-09-15 22:11:04

105

Views

Solution containing Riemann Zeta function for an integral involving the EGF of the Bernoulli/Euler polynomials

Published on 15 Sep 2021 - 22:11

score 103 · Answer 15 · 2021-09-27 22:45:07

103

Views

Fourier coefficients of Bernoulli periodic polynomials in terms of sine and cosine.

Published on 27 Sep 2021 - 22:45

List Question

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?

Forward difference of Bernoulli polynomials

Proving Property of Bernoulli's polynomial $B_n(x+1)-B_n(x)=nx^{n-1}$

Formula for alternating sum of odd powers of consecutive integers

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

Is it a well-known sequence of polynomials?

The alternating Fourier series associated with the fourth Bernoulli polynomial

What's the nicest proof of the formula for the sum of the $k$-th powers of the first natural numbers?

Dirichlet series of Bernoulli polynomials

Recurrence with Bernoulli-Barnes Polynomials

Elementary method to prove that $|\overline B_{2n}(x)|\le|B_{2n}|$?

Sum with Bernoulli poly. $\sum_{k=0}^{\lfloor n/2 \rfloor}\binom{n+1}{2k} B_{2k}(1/4) $

Bernoulli Polynomials and Euler summation

Solution containing Riemann Zeta function for an integral involving the EGF of the Bernoulli/Euler polynomials

Fourier coefficients of Bernoulli periodic polynomials in terms of sine and cosine.

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