Questions tagged [closed-form]

A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

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Solve and asymptotic expansion of $\sum_{a=1}^{H} \sum_{b=a+1}^{H} \left\lfloor{\frac{H}{a\, b}}\right\rfloor$

I am solving constrained polynomial systems resulting in constrained sums. I am looking to see if $$\sum_{a=1}^{H} \sum_{b=a+1}^{H} \left\lfloor{\frac{H}{a\, b}}\right\rfloor$$ is expressible in ...
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Integral $\int_0^1 \frac{\ln(1+x+x^2)\ln(1-x+x^2)}{x}dx$

Prove $$\sf I=\int_0^1 \frac{\ln(1+x+x^2)\ln(1-x+x^2)}{x}dx=\frac{\pi}{6\sqrt{3}}\psi_1\left(\frac{1}{3}\right)-\frac{\pi^3}{9\sqrt{3}}-\frac{19}{18}\zeta(3).$$ I have thought about the integral ...
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Closed form for $\int_0^t(x+c)^p(1-2x)^{N-1}\text{d}x$

I am trying to find a closed form for the integral $$I\equiv\int_0^t(x+c)^p(1-2x)^{N-1}\text{d}x,$$ where $N\in\mathbb{N}$, $p>0$, $c\ge 0$ and $t\in\left(0,\frac{1}{2}\right)$. I thought to ...
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Finding closed form of exponential generating function involving identity permutation

Fix a prime number $p > 1$ and for a positive integer $n$, let $a_n$ be the number of permutations $π ∈ S_n$ such that $π^p = id$, where $id$ is the identity permutation. Find a closed form for the ...
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