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0
votes
0answers
7 views

Rearranging a sequence to minimize a series function on its subsequences

Given an arbitrary nonempty sequence of integers $Q$ with $p$ elements: Let $R$ be a rearrangement of $Q$. Let $A$ be the solution set of $1\leq n\lt m\leq p$, where $n,m\in \mathbb{Z}$. $F(n,m) = ...
2
votes
3answers
323 views

How did the summation operation come into use? [on hold]

So we've been using summations at least since the dawn of calculus. I'm wondering how the process of summing a function came to be known? Are there events that led to the invention of the summation ...
2
votes
0answers
74 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 $ ...
-1
votes
2answers
61 views

Inner Product of Given Sum Positive Sequence

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in ...
-1
votes
1answer
92 views

Oddify an even function and vice versa: need a Fourier transform-based formula [closed]

I have constructed an operator that applied to an odd function will give its even counterpart and also an inverse operator that would transform an even function into an odd one. I need a ...
1
vote
1answer
155 views

Name for series $\sum f_n x^n / (n! (n+k)!)$

Let $(f_n)_{n\ge0}$ be a real sequence. Then $\sum f_n {x^n \over n!}$ is called the exponential generating function of $(f_n)$. Let $k\ge0$ be a nonnegative integer. If we add another factorial ...
6
votes
1answer
215 views

Is this series well known?

I recently encountered the following function $$ f(t) = \sum_{n=0}^\infty \frac{t^{n^2}}{n^2!}. $$ It seems familiar, though I cannot remember where I might have seen it before. I would like to know ...
1
vote
2answers
56 views

Summation mollifier to ensure a certain alternating series has the correct value

I would like the function $f(n,M)$, where $n$ and $M$ are integers and $n\le M$, so that $f$ satisfies the following two conditions: (1) $\sum_{n=0}^M (-1)^n f(n,M) = \tfrac{1}{2}$ (2) $f(n,M) ...
0
votes
0answers
46 views

$\eta(s)$ expressed as an 'alternating' sum of Hurwitz Zetas. Why does it only work for sums with an even number of terms?

It is known that: $$\zeta(s)= a^{-s}\,\sum_{k=1}^{a} \zeta_H\left(s,\frac{k}{a}\right)$$ is valid for all $a \in \mathbb{N}$ and all $s \in \mathbb{C}\,/1$, with $\zeta_H$ being the Hurwitz zeta ...
2
votes
2answers
311 views

Conjectured relation between alternating Prime zeta series and Riemann zeta

Let $P(s)$ be the Prime zeta function. Numerical evidence suggests these identities: $$ \sum_{k=1}^\infty \frac{(-1)^{k}P(3k)}{k}=\log{\bigg(\frac{1}{945}\frac{\pi^6}{\zeta(3)}\bigg)}\qquad\quad ...
1
vote
0answers
43 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 ...
0
votes
0answers
54 views

Where the following interpolation method converges?

In this question about discrete-analytic functions (that is functions, who equal to their Newton series) I asked for a solution for the following problem: Is there a method to extend the notion ...
7
votes
0answers
141 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 ...
0
votes
1answer
225 views

Prove that these two definitions of “natural” integration constant coincide when both converge

These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details). The first one is based on ...
4
votes
0answers
120 views

Without Skolem–Mahler–Lech Theorem? [closed]

Using Skolem–Mahler–Lech theorem one can easily prove the $\displaystyle \lim_{n\to +\infty}\left|\Re\left(\frac{1+i\sqrt{7}}{2} \right)^n\right| =+\infty$. Is there a "simple way" to prove this ...
3
votes
0answers
152 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 ...
0
votes
0answers
123 views

a question on sum of Gaussian binomial coefficients

I was trying to calculate something and at some point I get the following sum: \begin{equation} \sum_{t=0,t \text{ even}}^{s}{s+3n \brack s-t}\sum_{i = 0}^{t/2}q^{2i^2}{t/2+2n-i \brack t/2-i}{n ...
0
votes
2answers
137 views

Proof of equidistribution theorem for exponential coefficients

Can anyone provide a proof of the equidistribution theorem using Weyl's criterion for the case of $c*a \,\,\, \text{(mod 1)}$ where $c=2^n: \,\,\, \forall n \in N_0$ for irrational algebraic $a$? The ...
0
votes
1answer
73 views

Pointwise (in time) convergence in $H^{-1}$ implies pointwise weak convergence in $L^q$, why?

Let $u_n \to u$ in $C^0([0,T];H^{-1}(\Omega))$ and suppose $\lVert u_n \rVert_{L^\infty(0,T;L^\infty(\Omega))} \leq C$ for all $n$. It follows that for almost all $t$, $u_n(t)$ is bounded in ...
14
votes
1answer
392 views

A sum by Ramanujan for $\coth^{2}(5\pi)$

Ramanujan mentions in one of his letters to Hardy that $$\frac{1^{5}}{e^{2\pi} - 1}\cdot\frac{1}{2500 + 1^{4}} + \frac{2^{5}}{e^{4\pi} - 1}\cdot\frac{1}{2500 + 2^{4}} + \cdots = ...
2
votes
0answers
168 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 $n \in N \cup {0}$?
4
votes
0answers
153 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. ...
11
votes
1answer
250 views

Cantor set intersecting a geometric sequence

I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...
-4
votes
1answer
162 views

A general question on nonnegative integer sequence [closed]

Let $A=\{x\ |\ x\in\mathbb Z_{\ge 0},\ x\ $ with some conditions$\ \}$. Let $B=\mathbb Z_{\ge 0}-A$. Define $\ 2A= \{a+b : a \in A,\ b \in A\}$. Define $\ 2B=\{a+b : a \in B,\ b \in B\}$. Then the set ...
1
vote
0answers
52 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
1answer
81 views

Expression for infinite sum of two Bessel functions and a power

I'm searching for an expression of the following sum (all indices are integers and all variables are positive real numbers): $$\sum_{\lambda=-\infty}^{+\infty} A^{|l-\lambda|} B^{|\lambda|} ...
2
votes
1answer
112 views

How to classify the complex function with same natural boundary in complex plane? [closed]

There are complex functions with the same natural boundaries in the complex plane, but,they are different from each other. For example, there are lots of different lacunary power series with ...
3
votes
2answers
291 views

If two functions are equal to their Newton series, is their composition also equal to its Newton series?

Suppose we have two real-valued functions $f(x)$ and $g(x)$, both equal to their Newton series expansion: $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$ $$g(x) = \sum_{k=0}^\infty ...
8
votes
4answers
419 views

Computing $\int_0^T e^{itA}Be^{-itA} dt$ without an infinite series

I'm hoping to compute the following integral: $\int_0^T e^{itA}Be^{-itA} dt$ where $iA, iB$ are traceless anti-Hermitian matrices (i.e. $\mathfrak{su}(n)$). I have found the following form for the ...
5
votes
0answers
223 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 ...
3
votes
0answers
91 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 ...
4
votes
2answers
397 views

About a failed conjecture on $d(N)$

This was post by me on Maths SE: but it did not get any solution Some months ago I made the following conjecture - Let $d(n)$ denote the number of divisors of $n $. Then let $N$ be a number such ...
4
votes
1answer
279 views

A question on null sequences

Is it true that a sequence of real numbers $\{a_n\}$ converges to zero if and only if the sequences $\{\sin^2(nh)a_n\}$ $(h \in \mathbb{R})$ all converge to zero? In case the answer is ...
3
votes
2answers
359 views

Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers?

OEIS A226181: 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, ... Primes $p$ ...
12
votes
1answer
285 views

Cesaro(?)/Euler(?) - summation of the $s(p)=\sum_{k=0}^\infty (-1)^{H(k)} (1+k)^p$ for $p=1,2,3,…$ (where $H(k)$ is the Hamming-weight)

In another thread (in MO) there was a question about a series where the signs at the terms alternate with the "Hamming-weight", that means according to the number of bits in the binary representation ...
3
votes
2answers
345 views

Sum of series $a^{i^2}$

Is there any closed form known for the expression $\sum_{i=1}^\infty a^{i^2}$ where $|a|<1$? Thanks!
0
votes
1answer
314 views

Is a certain sumset derived from primes of a certain form the set of all naturals?

OEIS A167055 Numbers n such that $12n + 5$ is prime. $0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,...$ are items of OEIS $A167055$. I conjecture that the set of the sum of every two items of this ...
1
vote
2answers
247 views

What is known about $\displaystyle \sum_k{a^{b^k}}$?

What is known about $\displaystyle \sum_k{a^{b^k}}$? I am very interested in the possible applications of this series. I have asked about this on Mathematics Stack Exchange here. I'm wondering if ...
16
votes
2answers
353 views

Shortest supersequence of all permutations of $n$ elements

Given an alphabet with $n$ characters, what is the shortest sequence that contains all $n!$ permutations as subsequences? A subsequence can be obtained from a sequence by deleting any characters, ...
3
votes
1answer
246 views

Can the Fourier series of a continuous function diverge on an uncountable set of measure zero?

I know that there exist continuous function $f: [0,2\pi]\rightarrow\mathbb{R}$ whose Fourier series diverges at all rational points of $[0,2\pi]$(c.f. Katznelson).We also know that the set of ...
1
vote
0answers
115 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 ...
0
votes
1answer
80 views

A limit of a sum related to integer lattice and power series

I have the following lemma that I would like to find a source to cite for. Let $L$ be a subset of $\mathbb Z^d_{>0}$. I would like to claim that the limit $$\lim_{z \to (1,\ldots,1)^-} (\sum_{v ...
4
votes
0answers
157 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}} + ...
0
votes
2answers
94 views

Hypergeometric sum specific value

How to show? $${}_2F_1(1,1;\frac{1}{2}, \frac{1}{2}) = 2 + \frac{\pi}{2} $$ It numerically is very close, came up when evaluating: $$ \frac{1}{1} + \frac{1 \times 2}{1 \times 3} + \frac{1 \times 2 ...
1
vote
2answers
308 views

How to prove $\mathop {\lim }\limits_{x \to \infty } \sum\limits_{{f_x}(p) = 1} {\frac{1}{p}} = \ln 2$ for $p \le x$?

Let ${f_x}(m) = \sum\limits_{\left. p \right|m} {{f_x}(p)}$ be a strongly additive function on positive integer number $m$, where $p$ is a prime number. Set $${f_x}(p) = \left\{ ...
0
votes
0answers
19 views

Show that uniform continuity implies stochastic equicontinuity

Let $\Theta$ be a metric space and assume it is compact. Let $W_t: \Omega \rightarrow \mathbb{R}^k$ be a random variable for $t\leq T$. Let $m(.,\theta): \mathbb{R}^k\rightarrow\mathbb{R}^s$. Let ...
1
vote
2answers
196 views

Question on Morse inequalities

I want to understand why: From K.C Chang's book "Infinite Dimensional Morse Theory and Multiple Solution Problems": if i have then $(4.1)$ is formal : it means that EDIT1: $(4.1)$ tel us that ...
15
votes
1answer
575 views

Ramanujan's pi formulas with a twist

Given the binomial function $\binom{n}{k}$, define the following sequences, $$\begin{aligned} u_1(k) &= \tbinom{2k}{k}\tbinom{3k}{k}\tbinom{6k}{3k} = 1, 120, 83160, 81681600,\dots \\ u_2(k) ...
1
vote
2answers
212 views

Looking for a limit related to the series in a previous post

Can any one show that the following limit? $$ \lim_{z\rightarrow \infty} \sqrt{z} \: e^{-z}\sum_{k=1}^\infty \frac{z^k}{k! \sqrt{k}} \quad \stackrel{?}{=} \quad\sqrt{2}-1. $$ If one uses the ...
3
votes
2answers
344 views

How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration? $$ \int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0. $$ This post is related to my previous question here , ...