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 85 · Answer 1 · 2026-02-23 06:41:04

85

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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 23 Feb 2026 - 6:41

score 291 · Answer 2 · 2026-02-23 06:34:48

291

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Forward difference of Bernoulli polynomials

Published on 23 Feb 2026 - 6:34

score 251 · Answer 3 · 2026-02-23 06:40:54

251

Views

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

Published on 23 Feb 2026 - 6:40

score 196 · Answer 4 · 2026-02-23 06:39:34

196

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Formula for alternating sum of odd powers of consecutive integers

Published on 23 Feb 2026 - 6:39

score 101 · Answer 5 · 2026-02-23 06:41:05

101

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 23 Feb 2026 - 6:41

score 75 · Answer 6 · 2026-02-23 06:36:32

75

Views

Is it a well-known sequence of polynomials?

Published on 23 Feb 2026 - 6:36

score 144 · Answer 7 · 2026-02-23 06:39:33

144

Views

The alternating Fourier series associated with the fourth Bernoulli polynomial

Published on 23 Feb 2026 - 6:39

score 111 · Answer 8 · 2026-02-23 06:41:01

111

Views

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

Published on 23 Feb 2026 - 6:41

score 130 · Answer 9 · 2026-02-23 06:36:31

130

Views

Dirichlet series of Bernoulli polynomials

Published on 23 Feb 2026 - 6:36

score 48 · Answer 10 · 2026-02-23 06:34:49

48

Views

Recurrence with Bernoulli-Barnes Polynomials

Published on 23 Feb 2026 - 6:34

score 122 · Answer 11 · 2026-02-23 06:36:34

122

Views

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

Published on 23 Feb 2026 - 6:36

score 76 · Answer 12 · 2026-02-23 06:39:39

76

Views

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

Published on 23 Feb 2026 - 6:39

score 86 · Answer 13 · 2026-02-23 06:34:33

86

Views

Bernoulli Polynomials and Euler summation

Published on 23 Feb 2026 - 6:34

score 106 · Answer 14 · 2026-02-23 06:38:01

106

Views

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

Published on 23 Feb 2026 - 6:38

score 104 · Answer 15 · 2026-02-23 06:40:51

104

Views

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

Published on 23 Feb 2026 - 6:40

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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