Questions tagged [infinite-sequences]

The tag has no usage guidance.

21 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
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(-...
Greg Egan's user avatar
  • 2,852
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}(\...
Goulifet's user avatar
  • 2,174
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 ...
Matteo Beccaria's user avatar
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 ...
Darren Ong's user avatar
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):=\{...
Mare's user avatar
  • 25.8k
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. ...
the_sandcastler's user avatar
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 ...
popa13's user avatar
  • 31
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 ...
Nikita Kalinin's user avatar
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 ...
Christopher D. Long's user avatar
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-...
Gabriel Medina's user avatar
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}{...
John Finkelstein's user avatar
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\;\...
pie's user avatar
  • 139
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 $...
jrranalyst's user avatar
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(\...
user35593's user avatar
  • 2,286
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 $f\in\mathbb{N}^\mathbb{N}$ ...
Manlio's user avatar
  • 302
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 ...
Stefano Luzzatto's user avatar
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 ...
dspyz's user avatar
  • 263
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 ...
CfourPiO's user avatar
  • 159
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\...
Sayan Dutta's user avatar
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 ...
Corbin's user avatar
  • 424
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 ...
AIGuy's user avatar
  • 1