All Questions
9 questions
6
votes
1
answer
283
views
On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence
In a previous MO post, H. Cohen suggested Gorodetsky's 2021 paper which discussed $6+6+3=15$ "sporadic sequences". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-...
- 12.6k
2
votes
0
answers
159
views
Finding similar Zudilin-Cohen recurrence relations and cfracs for $\frac{\zeta(4)}{13}$?
I. Two recurrence relations
The first one was also discussed in this MO post. We have the similar,
\begin{align}
(n+1)^5 u_{n+1} &= (2n + 1)(9n^2 + 9n + 3)(15n^2 + 15n + 4)u_n +3n^3(9n^2-1)u_{n-1}\...
- 12.6k
35
votes
2
answers
1k
views
Representations of $\zeta(3)$ as continued fractions involving cubic polynomials
$\zeta(3)$ has at least two well-known representations of the form $$\zeta(3)=\cfrac{k}{p(1) - \cfrac{1^6}{p(2)- \cfrac{2^6}{ p(3)- \cfrac{3^6}{p(4)-\ddots } }}},$$
where $k\in\mathbb Q$ and $p$ is a ...
- 13.4k
0
votes
0
answers
356
views
Is $\lim_{x\to\infty}\sum_{n=-\infty}^{\infty}\frac{x}{n^2+x^2}=\pi$?
It seems that
$$\lim_{x\to\infty}\sum_{n=-\infty}^{\infty}\frac{x}{n^2+x^2}=\pi.$$
But I can't prove it. I cannot prove that the function is decreasing in $x$ either.
- 5,103
11
votes
2
answers
882
views
Do infinitely nested radicals have any applications?
There is a simple necessary and sufficient condition for a continued radical of the form $\sqrt{a_1 + \sqrt{a_2 + \dotsc}}$ to converge (where all terms $a_1, a_2$ etc. are nonnegative). Namely, that ...
- 4,943
9
votes
5
answers
2k
views
Laplace's summation formula
I recently came across the following formula, which is apparently known as Laplace's summation formula:
$$\int_a^b f(x) dx = \sum_{k=a}^{b-1} f(k) + \frac{1}{2} \left(f(b) - f(a)\right) - \frac{1}{...
- 3,283
8
votes
0
answers
237
views
Sequences for which $\prod (1-z^n)^{a(n)}$ is a polynomial
This is mostly a reference request.
I'm working with complex coefficients, although all I have in mind have integer coefficients.
Let $a=(a(n))_{n\ge 1}$ be a sequence, say of integers (I have non-...
- 63.9k
2
votes
2
answers
655
views
Expansion of inverse logarithmic integral in terms of lambert w
Cipolla and Césaro both gave expansions of $\operatorname{li}^{-1}$ in tems of nested $\log$ functions. I think it can be written in terms of the Lambert-W function in the form:
$\operatorname{li}^{-...
- 1,903
2
votes
0
answers
222
views
Four kinds of generalized hypergeometric formulas for $\pi$
Given,
$$\begin{array}{|c|c|c|c|}
\hline
n&a_n&b_n&c_n\\
\hline
1 & 6541681608 & 163096908 & -640320^3\\
\hline
2 & 85840 & 4492 & -14112^2\\
\hline
3 & 28302 &...
- 12.6k