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14
votes
0answers
810 views

Regularizing the divergent sum $1^k + 2^k + \cdots$

EDIT: Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$ I was looking at ...
12
votes
0answers
523 views

How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's answer to the question Freeness of a Z[x] module depicted below? -- Choosing colors according to the highest power of 2 which ...
8
votes
0answers
467 views

Can an infinite sum depending on the logarithms of all positive integers be rational or algebraic?

Consider $$ C = \sum_{n=1}^\infty \frac{(-1)^{f_1(n)} f_2(n) \log{n} + f_3(n)}{f_4(n)}$$ The sum converges. $f_1$ is either $0$ or $n-1$, $f_2,f_3,f_4$ are polynomials with integer coefficients and $ ...
8
votes
0answers
377 views

Composition of two formal series

There are two formal semi-infinite Laurent series $$ f_+(z)=z+\sum_{k=2}^{\infty} a_k z^k $$ and $$ f_-(z)=z+\sum_{k=0}^{\infty} b_k z^{-k} $$ Their composition (we assume that this composition ...
7
votes
0answers
156 views

If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
6
votes
0answers
175 views

Irrationality of the sum of the reciprocal of perfect powers

A couple of days ago I was trying to remember a classical exercise (which I now find out goes by the name of Goldbach-Euler theorem). Eventually I figured out that it asked to prove that ...
6
votes
0answers
209 views

Zeros of polynomials related to Jensen polynomial associated with Riemann xi function $\xi(x)$

We encountered polynomials defined by the recursive relations for the coefficients $b_k>0$ as defined below: $$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$ ...
6
votes
0answers
330 views

Graphs with graphic imbalance sequences

Let $G$ be simple undirected graph and $e=uv\in E(G)$. The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$. Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of ...
5
votes
0answers
238 views

On the sum of consecutive primes and product of first and last

Lets consider the sequence of consecutive prime numbers $p_1=2 , p_2=3 ,p_4=5 , \cdots$ . $(p_n,p_{j})$ is to be called good prime pair if $$\sum_{i =n }^{j}p_i= p_n p_{j}$$ Meaning the sum of set of ...
5
votes
0answers
122 views

Inverse problems for an asymptotic series which depends on a parameter?

I have the series $\sum_{n=0}^{\infty}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$, where $m$ is an integer. Is it possible to compute the coeffients $a_{n}(\nu)$? An ...
5
votes
0answers
344 views

Number of Configurations in the optimal Hanoi tower

There is a unique strategy how to move $n$ disks from the first rod to the second optimally and it takes $2^n-1$ steps, solution is obtained by simple recursion. I am interested into the following ...
5
votes
0answers
338 views

Does every strictly increasing, unbounded sequence of positive real numbers contain arbitrarily long, finite subsequences which are “sort of increasing” or “sort of decreasing” (as defined below)?

Is the following true? If $(x_0, x_1, \dots)$ is a strictly increasing, unbounded sequence of positive real numbers, then there exist fixed $M,N \geq 1$ such that the sequence $(x_0, x_1, \dots)$ ...
4
votes
0answers
156 views

a question about Tsirelson's space

NOTE: I asked this question over at math.stackexchange.com but got no answer or comments after 3 days, probably because it's a bit specialized. Hopefully it is interesting enough to ask over here. ...
4
votes
0answers
179 views

Ramanujan's series for $(1/\pi)$ and modular equation of degree $29$

In his famous paper "Modular Equations and Approximations to $\pi$", Ramanujan gives the following famous series for $1/\pi$: $$\frac{1}{2\pi\sqrt{2}} = \frac{1103}{99^{2}} + ...
4
votes
0answers
213 views

$\sum_{p,q \text{ primes } p \le q} 1/(pq\log(pq))$

The sum $$ \sum\limits_{p,q \text{ primes } p \le q} \frac{1}{pq\log(pq)}$$ is related to a conjecture of Erdős about primitive sequences. It converges because the sequence is primitive. If my ...
4
votes
0answers
230 views

Useful lower bound on an infinite sum

Fix integer $s.$ I have encountered the following infinite sum. $$\sum_{k=0}^\infty \Pi_{l=1}^k\left[1-(1-2^{-l})^s\right]$$ Is there a useful lower bound on this expression? For instance, if $s=1,$ ...
4
votes
0answers
152 views

Are there infinite sequences of rational cubes whose first differences are positive squares?

This is related to How many sequences of rational squares are there, all of whose differences are also rational squares? Are there infinite sequences $a_n$ of rational cubes whose first ...
4
votes
0answers
197 views

Number of times lead changes in a multi-candidate election (reference-request)

In a two candidate election where votes are distributed uniformly at random between the candidates, the probability that the lead changes when tallying the $i$-th vote is the same as the probability ...
4
votes
0answers
682 views

Computability of OEIS A034891 …partitions of n into prime parts (1 included)

On the seqfan mailing list RGWv gave short algorithm for computing A000041 number of partitions of n the partition numbers: f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == ...
3
votes
0answers
161 views

Do $r$-th root Harmonic numbers ever sum to integers?

None of the Harmonic numbers $H_n = \sum_{k=1}^n 1/k$ are integers for $n>1$ (e.g., this MSE question and answer). Q. Define the $r$-th root Harmonic number $H_n^{1/r} = \sum_{k=1}^n ...
3
votes
0answers
92 views

On the comparison of Egyptian fractions of two kinds

I posted the question on MSE here but it did not get any answer. Consider $$S(n)=\left\{(a_1 ,a_2,a_3, \dots, a_n)\mid a_1\le a_2\le\cdots\le a_n, \; \sum_{r=1}^{n}\frac{1}{a_r} = 1\right\} \subset ...
3
votes
0answers
106 views

“Shifted” Vandermonde determinant is nonzero?

I have already posted this question at MSE here, but as it received a few upvotes, but no comments or answers I choose to cross-post it here. Let $P$ be a degree-two polynomial, with roots ...
3
votes
0answers
109 views

Prove that when converge, the following expansions are equal

Prove $f_1(x)=f_2(x)=f_3(x)$ when converge. $$f_1(x)=\sum_{m=0}^{\infty} \binom {x}m \sum_{k=0}^m\binom mk(-1)^{m-k}f(k)$$ $$f_2(x)=\lim_{n\to\infty}\binom xn\sum_{k=0}^n\frac{x-n}{x-k}\binom ...
3
votes
0answers
246 views

The functional equation of Hofstadter's Q sequence

Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and $Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything on this sequence has been proved -- not even that $Q(n)$ is ...
3
votes
0answers
140 views

Shift-invariant submultiplicative seminorms of $\ell^{\infty}$

Question: Is there a shift-invariant submultiplicative seminorm $||\cdot||$ of $\ell^\infty$ which satisfies the following property? If $f:\mathbb{N}\rightarrow\mathbb{N}$ is an increasing function ...
3
votes
0answers
260 views

Convolution inverse of recursively defined sequence is alternating

Consider the double sequence $A(n,k)$ which is recursively defined by $$A(n,n)=1 \text{ for } n=0,1,2,\dots \text{ and }$$ $$A(n,k)=2\sum_{l=1}^{k+1} \binom{2n+1}{2l} A(n-l,k+1-l) \text{ for }0\leq k ...
3
votes
0answers
220 views

Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k $ at $\small x=-1$)

I'm still fiddling with this recent question and come to a detail, whether I can find closed forms for the sums of the $\small lngamma() $ function. Precisely $$ \small f_p(x) = \sum_{k=0}^{\infty} ...
3
votes
0answers
689 views

Method for variable substitution in multiple summation

I want to ask: is there any general method for variable substitution in multiple summation? For example in the following equation a new variable $\lambda=n+m-2\mu$ is introduced to transform the LHS ...
3
votes
0answers
327 views

Finch's sequence over $\mathbb{F}_3$

In http://algo.inria.fr/csolve/seqmod3.pdf -- "Periodicity in sequences mod 3" Steven Finch (also cited in Sloane's OEIS A112683) defines the following sequences in $\mathbb{F}_3$: For each positive ...
2
votes
0answers
186 views

Minimizing $\{0,1\}$-sequence permutations

Explanation: For a given bit sequence $f$, reposition the bits as to minimize $G$ which can be thought of as a measure of how poorly proportional $f$ is to each of its subsequences. Let $p \in ...
2
votes
0answers
99 views

The behavior of series involving special subsets of the prime numbers

It is well known that the series $\sum_{p\in \mathbb{P}} \frac{1}{p}$ diverges where $\mathbb{P}$ denotes the set of primes. Brun proved that $\sum_{p\in \mathbb{P_2}} \frac{1}{p}$ converges where $ ...
2
votes
0answers
182 views

closed form expression for an infinite series

Is there any closed form expression for the infinite sum $\sum_{n \geq 0}q^{n(n+1)/2}(1+q)(1+q^2)\cdots(1+q^n)u^n$ where both $q$ and $n$ are variables and $n \in N \cup {0}$?
2
votes
0answers
109 views

A second polylogarithm ladder for the tribonacci and n-nacci constants

In "A seventeenth-order polylogarithm ladder", on page 6 (eq 25), Bailey et al give a dilogarithmic ladder that begins with, $$0 = ...
2
votes
0answers
128 views

Series for envelope of triangle area bisectors

The lines which bisect the area of a triangle form an envelope as shown in this picture It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is ...
2
votes
0answers
191 views

Proving that an increasing iterative sequence increases at a decreasing rate

In this question Proving a sequence of integrals increases (iterated minimax distributions) Pietro Majer proved that $$\int_0^1F_n(x) dx \leq \int_0^1F_{n+1}(x) dx$$ when $$F_n(x) = ...
2
votes
0answers
406 views

How to calculate/approximate expectation of function of a binomial random variable?

Hi, I am stuck at following problem in my research. Suppose that $M=m$ is a random variable with binomial distribution with parameters $n,p$. The constants $r$ and $\gamma$ are greater than zero. ...
2
votes
0answers
519 views

What square-summable sequences are “sinc-summable”?

$\operatorname{sinc} : \mathbb{R} \to \mathbb{R} \;\;$ is defined by $\;\; \operatorname{sinc}(x) \; = \; \begin{cases} 1 & \text{if }\:\;x=0 \\ \\ \frac{\operatorname{sin}(x)}x & \text{else} ...
2
votes
0answers
476 views

recursive sum of products of bessel functions

I have found a way to sum products of bessel functions in the form $$S_\ell(x,y)=\sum_{n=-\infty}^\infty (-1)^{n+\ell} I_{\ell-2n}(x)I_n(y)$$ recursively, i.e. once $S_0(x,y)$ is found, via the ...
1
vote
0answers
46 views

Algebraic Relations between $G(q)$ and $H(q)$

I had posted this originally on MSE, but got no response at all hence posting the same here. In his paper "Algebraic Relations between Certain Infinite Products" (Proceedings of the London ...
1
vote
0answers
67 views

Convergence and summation of power towers or Ackermann functions

I deleted this at StackExchange as it seems advanced or not very relevant for there and I want to ask here. Let z be a complex number, $Im(z)\neq 0$ or $Re(z)<4$, m be a natural number, let ...
1
vote
0answers
123 views

Reference for a General Theory of Sequences?

Since decades, mathematicians are studying function spaces, discovering new structures more and more adapted for a general theory of functional analysis. In that works, sequence spaces are generally ...
1
vote
0answers
103 views

Passing to the limit in a PDE (subsequence problems)

For $w \in L^2(0,T;H^1)$, consider the PDE $$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$ where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and ...
1
vote
0answers
285 views

Do these infinite series expressing $\zeta(s)$ only (partially) converge at $\Re(s)=\frac12$?

The following analytic continuation for $\zeta(s)$ towards $\Re(s)>-1$ was derived here: $$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(\sum _{n=1}^{\infty } {\frac {s-1-2\,n}{{n}^{s}}} + ...
1
vote
0answers
86 views

A generalization of alternating series involving modulus?

Alternating series are common in the literature, with important examples including $\displaystyle\sum_{n=1}\frac{(-1)^{n-1}}{n}=\log 2$, ...
1
vote
0answers
90 views

Structural differences between closed forms of two related infinite products?

In this question, I went quite a bit over the top, so I now tried to rephrase it in a much simpler way. Take $a \in \mathbb{R}, s \in \mathbb{C}$ and: $$\displaystyle C(s,a) := \prod_{n=1}^\infty ...
1
vote
0answers
203 views

Convergence of $\sum_{n=1}^\infty \frac{\psi^{(1)}(1-n/\pi)}{n^3}$

Related to an open problem about another series. Set $$A= \sum_{n=1}^\infty \frac{\psi^{(1)}(1-n/\pi)}{n^3}$$ where $\psi^{(n)}(k)$ is the polygamma function. Does $A$ converge? The related ...
1
vote
0answers
78 views

The $d$-dimension extension of Bernoulli Polynomial

It is known that Bernoulli polynomial has the following Fourier expansion: \begin{equation*} B_{2n}(x) = \frac{(-1)^{n-1}2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos(2k\pi x)}{k^{2n}}. ...
1
vote
0answers
206 views

Zeta sum $\sum_{n=2}^\infty \frac{\zeta(n)}{a^n}$

Probably this is known, but mathworld and wolfram alpha don't recognize this potential identities. Numerical evidence suggests: $$ \sum_{n=2}^\infty \frac{\zeta(n)}{a^n} =? \sum_{n=1}^\infty ...
1
vote
0answers
110 views

Bounding a recursively defined sequence

I have a sequence $\lambda_0,\lambda_1,\ldots,$ which is defined recursively as $$ \lambda_0 = \frac{1}{2},$$ and $$\lambda_{k+1} = \max_{\lambda\in [1,b]} \left(\frac{1}{2\lambda}\prod_{0\leq ...
1
vote
0answers
163 views

Calculating $n$ for $\sigma(\sigma(n)-n) = n$ [redefined]

As in A072868 described by OEIS; Defined by $\sigma(\sigma(n)-n) = n$. Since these numbers are important in regard to many things, specially mersenne primes, since ${n-1 \over 2}\times ...