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4
questions
2
votes
1
answer
164
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Another weird limit involving gamma and digamma function via continued fraction
Context :
I want to find a closed form to :
$$\lim_{x\to 0}\left(\frac{f(x)}{f(0)}\right)^{\frac{1}{x}}=L,f(x)=\left(\frac{1}{1+x}\right)!×\left(\frac{1}{1+\frac{1}{1+x}}\right)!\cdots$$
Some ...
1
vote
0
answers
52
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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
9
votes
2
answers
341
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On the cubic counterpart of Ramanujan's $\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots$?
We have Ramanujan's well-known,
$$\sqrt{\frac{\pi\,e}{2}}
=1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots\color{blue}+\,\cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\...
- 54.9k
3
votes
0
answers
160
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Two complementary continued fractions that are algebraic numbers
Define the two similar continued fractions,
$$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} {3km\color{blue}+\cfrac{(2m-1)(2m+1)}{5km\color{blue}+\cfrac{(3m-1)(3m+1)}{7km\color{blue}+\ddots}}}}\tag1$$...
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