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3
questions
1
vote
0
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Does the rest of this family of continued fractions have closed forms?
The pattern for the continued fractions below is quite straightforward. $F_1$ has numerators with all the integers but,
$F_2\; \text{is missing}\; 2m+1 = 3,5,7,\dots\\
F_3\; \text{is missing}\; 3m+1 = ...
- 54.9k
3
votes
0
answers
173
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Question on a closed-form expression related to the harmonic number $H_n$
In this question the notation $\tilde{f}(x)$ refers to an analytic representation of the summatory function
$$f(x)=\sum\limits_{n=1}^x a(n)\tag{1}$$
that converges to
$$\underset{\epsilon\to 0}{\text{...
- 7,631
13
votes
3
answers
537
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Is $\frac{1}{\pi}\int_{0}^{\infty} \Gamma(\sigma +xi)^2\,\Gamma(\sigma-xi)^2 \,dx = \frac{\Gamma(2\,\sigma)^4}{\Gamma(4\,\sigma)}$?
Using the approach from the answer to this question, it can be shown that for $\sigma \in \mathbb{C}, x\in \mathbb{R}$:
$$\frac{1}{\pi}\int_{0}^{\infty} \Gamma(\sigma +xi)\,\Gamma(\sigma-xi) \,dx =\...
- 3,191