# Questions tagged [bernoulli-numbers]

Questions on Bernoulli numbers, a special sequence of rational numbers that arise as the coefficients in the power series expansions of certain elementary functions.

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### Asymptotically similar functions with opposite parity, were they considered, are they useful? Case of polynomials

So, can we transform an even function into an odd function and vice versa? Let's consider this method: Transformation even->odd: Suppose $f_{even}(x)$ is a function which satisfies the following ...
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### Use Bernoulli trials to select random k-length permutations

Using only Bernoulli trials with $p=0.5$, how can I efficiently create a permutation of length $k$ from a set of length $n$? I can think of a few ways to do this that waste trials, but I’d like to do ...
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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 $m$ ...
### How is $B_n = 1- \sum_{k=0}^{n-1} \binom{n}{k} \frac{B_k}{n-k+1}$ Where $B_n$ are the Bernoulli Numbers with $B_1 = \frac{1}{2}$
So I was browing Wikipedia just looking at Bernoulli Number identities and I stumbled across this $$B_m^+ = 1- \sum_{k=0}^{m-1} \binom{m}{k} \frac{B_k^+}{m-k+1}$$ The Wikipedia page said that this ...