Questions tagged [bernoulli-numbers]
The Bernoulli numbers are the rational numbers $B_n$ defined as the coefficients in the expansion $\frac{x}{e^x-1} = \sum_{n \geq 0} B_n \frac{x^n}{n!}$. They vanish when $n$ is odd and greater than $2$. They appear in the values at integers of the Riemann $\zeta$ function. These classical numbers play an important role in number theory and in several other places in mathematics.
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A question about generalized harmonic numbers modulo $p$
Let $p \equiv 1 \pmod{3}$ be a prime and denote $H_{n,m} = \sum_{k = 1}^n 1/k^m$ as the $n,m$-th generalized harmonic number. I'm interested in computing $H_{(p-1)/3,\, 2}$ and $H_{(p-1)/6,\,2}$ ...
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Ask for a reference or a proof of an identity involving a finite sum and the Bernoulli numbers
Let $B_{n}$ for $n\ge0$ denote the Bernoulli number generated by
\begin{equation*}
\frac{z}{\textrm{e}^z-1}=\sum_{n=0}^\infty B_n\frac{z^n}{n!}=1-\frac{z}2+\sum_{n=1}^\infty B_{2n}\frac{z^{2n}}{(2n)!},...
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Claim on divisibility of a power sum
Let $x,y,z$ are integer and $x,y>0$
Define $S(x,y)=1^y+2^y+3^y+...+x^y$
Can it be shown that
If given $z\ne0$ then $S(x,y)\equiv z\pmod{x}$ have finitely many solution of $x$ with respect to $y$.
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Simplifying a rational function in terms of Bernoulli numbers and polynomials
Faulhaber's formula expresses a sum over some finite number of naturals to the $m^{th}$ power in terms of the Bernoulli numbers $B_{j}$ (using the $B_{1} = 1/2$ convention) or polynomials $\hat{B}_{j}$...
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Ask for a proof of an inequality involving the Bernoulli numbers
Let $B_k$ be the Bernoulli numbers and let
\begin{equation}
T_k=\frac{2^{2k}}{(2k)!}|B_{2k}|, \quad k\ge1.
\end{equation}
Prove the inequality
\begin{equation*}
\frac{\frac{1}{k+2}\sum_{j=0}^{k+1}\...
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Probability of the maximum of a throw of an infinite number of $n$-sided dice being $k$ [closed]
Let $X$ be the random variable obtained as the maximum of a throw of $m$ dice (each of which is $n$-sided). In other words, $X = \max\{l_1,\cdots, l_m\}$ where $l_i$ can take any value between $1$ and ...
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Nicely motivated papers or book chapters on the formula for the sum of the $k$-th powers of the first natural numbers [closed]
Do you know of a text where I can find a nicely motivated proof of the formula for $1^{k}+2^{k}+\cdots+n^{k}$?
At the very beginning of page 68 of Professor H. S. Wilf's generatingfunctionology, one ...
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Is there a prime factor that in the numerator of Bernoulli number $B_{2n}$ strictly greater than $2n$ for any $n>7$?
As well known, Bernoulli number $B_{2n}$ is a rational number which can always be shown as $B_{2n}=V_{2n}/A_{2n}$ , where $\gcd(V_{2n}, A_{2n})=1$.
For example, $B_{16}=\frac{3617}{-510}$ and $3617$ ...
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Anti-concentration for Bernoulli summation
Suppose $\{ Y_i\}_{i = 1}^n$ is i.i.d. Bernoulli distribution with mean $p$. Denote the sample of $\{ Y_i\}_{i = 1}^n$ as $\overline{Y} = \frac{1}{n} \sum_{i = 1}^n Y_i$. I want to know where there ...
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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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