Questions tagged [infinite-sequences]
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21
questions with no upvoted or accepted answers
9
votes
0
answers
397
views
Can this infinite sum for the Riemann zeta function be generalised?
I recently derived the following identity (which is probably a rediscovery of something well-known to experts).
$$\sum_{k=1}^\infty{\frac{(-1)^{k+1}k^{4n+1}}{e^{k \pi}-(-1)^k}}=\frac{1}{2}\zeta\left(-...
7
votes
0
answers
133
views
Characterization of tempered distributions from tempered sequences
Let $\mathcal{D}(\mathbb{R})$ be the space of compactly supported and infinitely smooth functions for its usual topology. Let
$\mathcal{D}'(\mathbb{R})$ be the topological dual of $\mathcal{D}(\...
6
votes
0
answers
183
views
Sum of squared hypergeometric polynomials
$\sum_{m=1}^\infty \frac{1}{m} \bigg[{}_2F_1(-m,m,2,u)\bigg]^2 = \frac 1 4 -\frac 1 2 \log u$
I have very strong numerical support that this is true when
$0<u\le1$.
Can anyone help proving or ...
4
votes
0
answers
67
views
Draw an arbitrary line on a Penrose tiling. Determine a sequence of tiles can it intersect
Let us consider a Penrose tiling of $\mathbb R^2$. Starting with an arbitrary point on the tiling, draw an arbitrary straight line. Assume that this straight line never overlaps perfectly with a ...
4
votes
0
answers
131
views
Irreducibility of polynomials corresponding to sequences
I have no experience with this, so I dont know if this is too easy for MO.
Let $(a_n)$ be a strictly monotone sequence of natural numbers, then define the set of nice numbers of $(a_n)$ as $X(a_n):=\{...
3
votes
0
answers
179
views
Shift Operators and the Weyl Algebra
I have a question about the action of a shift operator $E$ on polynomials $Ep(x) = p(x+1)$ in the context of linear differential operators in one variable with polynomial coefficients, i.e. ...
3
votes
0
answers
61
views
Some exercise on the regularity of a summability method
I was reading the book of Johann Boos "Classical and modern method in summability theory" and I came across an exercise from the Chapter 2 (page 50, exercise 2.3.15). Here is the statement ...
2
votes
0
answers
59
views
Factor group of all the sequences by the subgroup of bounded sequences
Consider the group G of the sequences of real numbers (the group operation is addition). It contains a subgroup H of bounded sequences.
Is there any nice description of the factor group G/H ?
It is ...
2
votes
0
answers
196
views
Uniform distribution of log(log((n!)!)) mod 1
Can it be shown that the sequence $\log(\log((n!)!))$ is uniformly distributed mod 1? I believe it should be, but I'm not certain if Stirling's approximation repeated twice combined with the ...
2
votes
0
answers
235
views
Cardinal numbers and the Bolzano-Weierstrass theorem
Let $\kappa$ be a cardinal number, define $\textsf{M}(\kappa)$ and $\textsf{BW}(\kappa)$ as follows:
$\textsf{M}(\kappa)$ : For every sequence $(f_{n}:\kappa\to \mathbb{R})_{n\in\mathbb{N}}$ of real-...
2
votes
0
answers
1k
views
Is there an infinite product like this for $\cos x$?
There are infinite products of iterated square roots for $\log x$ and $\arccos x$ as functions of $x$. For example
$$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...
1
vote
0
answers
220
views
Is $\sum\limits_{k=1}^nk^m=S_3(n)\cdot\dfrac{P_{m-3}(n)}{N_m}$ for odd $m>1,\sum\limits_{k=1}^nk^m=S_2(n)\cdot\dfrac{P_{m-2}'(n)}{N_m}$ for even $m$?
I asked this question here
When I was in high school, I was fascinated by
$$
\sum\limits_{k=1}^n k= \frac{n(n+1)}{2}
$$ so I tried to find the general value of the sum
$$
\sum\limits_{k=1}^n k^m\;\...
1
vote
0
answers
146
views
Infinite matrices from $\ell^p$ to $\ell^{p/(p-1)}$ that are compact operators
I wanted to ask if my proof (sketch) of the following statement is correct. Namely, let $p>1$ and define $q= \frac{p}{p-1}$ we are given an operator $K : \ell^{p} \rightarrow \ell^{q}$ defined as $...
1
vote
0
answers
97
views
Generalizion of Euler identity with infinite sum of inverse squares
For $x,y\in \mathbb{R}\backslash \mathbb{Z}$ let
$$
f(x,y)=\sum_{n\in\mathbb{Z}} \frac{1}{(n-x)(n-y)}
$$
Is there a closed formula for $f(x,y)$?
What is known:
We have
$$
f(x,x)=\left(\frac{\pi}{\sin(\...
1
vote
0
answers
92
views
Existence of limit computable map
Is there a limit computable function $\Phi$ with the following properties?
Let $T\subset \mathbb{N}^{<\mathbb{N}}$ be a tree (coded as its characteristic function) and $f\in\mathbb{N}^\mathbb{N}$ ...
1
vote
0
answers
36
views
Positive density of certain arithmetic sequence
Let $\{a_i\}_{i=0}^{\infty}$ be a sequence of positive real numbers bounded above by some uniform constant. For each $k,n\geq 1$ let
$$
A^{(k)}_n:=\prod_{i=k}^{n+k-1}a_i
$$
and suppose that there ...
1
vote
0
answers
434
views
Greedy sequences without k-term arithmetic progressions
If $S_k$ is the greedy sequence with no length-k arithmetic subsequence, (ie $S_3$ = A003278 , $S_4$ = A005837 , $S_5$ = A020655 ), is it guaranteed that any other sequence $a$ with no length-k ...
0
votes
0
answers
41
views
Summation of the following form with non-integer n
I have the following function:
$$ G(z) = \sum_{j = 0}^{\infty} \frac{\Gamma(1 + n) z^j}{j ! (\Lambda + jC)^{n+1}} $$
If $n \quad \epsilon \quad \mathbb{Z}^{+} $, the above function can be ...
0
votes
0
answers
136
views
Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?
This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...
0
votes
0
answers
196
views
What are the hidden assumptions behind Harvey Friedman's claim, CSR?
I'm doing some archeology and trying to understand a claim. As summed up by David Roberts, on the FOM list in 2011:
Let the statement "every infinite sequence of rationals in [0,1] has an ...
0
votes
0
answers
2k
views
In a network with N nodes, what is the general formula for computing the propagation of a set of numbers?
I am creating a circular neural network with N nodes. Each node is connected via a send pathway to every other node, and the connection between two nodes has a weight. Any number sent over the ...