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-4
votes
1answer
178 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
92 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
286 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
124 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
317 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
522 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 ...
6
votes
0answers
271 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
99 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
410 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
303 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
373 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
335 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
391 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
323 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
266 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 ...
18
votes
2answers
486 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
292 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
134 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
85 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
1answer
429 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 $1/\pi$: \begin{align}\frac{1}{2\pi\sqrt{2}} &= \frac{1103}{99^{2}} + ...
0
votes
3answers
152 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
349 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\{ ...
1
vote
2answers
213 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 ...
17
votes
1answer
921 views

Ramanujan's pi formulas with a twist

Given the binomial function $\binom{n}{k}$. 1. 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
222 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
391 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 , ...
17
votes
3answers
2k views

Where in mathematics do these polynomials appear?

Does anyone recognize the following sequence of polynomials? $f_0(x) = x-1$ $f_1(x) = x^2-x$ $f_2(x) = x^4-2x^2+x$ $f_3(x) = x^8-3x^4+3x^2-x$ $f_4(x) = x^{16}-4x^8+6x^4-4x^2+x$ $\vdots$ The ...
-1
votes
1answer
258 views

Does anyone recognize this generating function [closed]

$a_1=1, a_2=1, a_3=3, a_4=15, a_5=105$ Reccurence formula is $a_{k+1}=\sum\limits_{\lambda_1+\lambda_2+\ldots+\lambda_s=k,\ \lambda_i\geq1} a_{\lambda_1}a_{\lambda_2}...a_{\lambda_s}{k \choose ...
9
votes
1answer
904 views

Has anyone seen this series?

I come across the following infinite series. $$ \sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}. $$ In particular, I am interested in the case where $a=1/4$. ...
1
vote
0answers
108 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 ...
-2
votes
1answer
84 views

Monotonic sequence (edited) [closed]

For any two n-dim vector $v$ and $v'$ define $v\leq v'$ iff for each $1\leq i\leq n$, $v_i\leq v_i'$. Suppose further that the entry of vectors can only take values from $m$ distinct values $\{a_1, ...
3
votes
0answers
117 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 ...
2
votes
3answers
452 views

Does this 'alternating' Euler product converge for all $\Re(s) > 0$?

Does the following 'alternating' Euler product, with $p_n$ the $n$-th prime number, converge for $\Re(s)>0$ ? $$\displaystyle \prod_{n=1}^\infty \left( \dfrac{1}{1-\frac{1}{p_{n}^{s}}} ...
3
votes
1answer
215 views

Collatz dropping times aperiodicity

Let $a(n)$ for $n \geq 1$ be the number of powers of two $2^m$ that lie between $3^{n-1}$ and $3^{n}$: $$a(n) = 1, 2, 1, 2, 1, 2, 2, 1, ...$$ It represents the increments between successive terms of ...
6
votes
0answers
212 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 ...
3
votes
0answers
112 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 ...
2
votes
1answer
137 views

a second order difference equation related to a real polynomials which seems to have only real roots

I am seeking solutions to the following difference equation: $$2c_k-c_{k-1}-c_{k+1}=\ln(k+A)-\ln(k+B)$$ where $A>B>0$. This equation is related to a real polynomial (see here) which I want to ...
6
votes
0answers
243 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: $$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$ ...
0
votes
0answers
166 views

Usage of multinomial theorem with infinite series

What are the conditions for using the multinomial theorem with infinite series? I have an expression but I don't know if I can use it. The expression is: $$ \left[\sum_{m=0}^{\infty} \frac{\mu^m ...
7
votes
3answers
423 views

Summation of a series

I would like to sum the series $$ \sum_{n=0}^\infty \frac{1}{(1+a^2 (n+1/2)^2) ^{3/2}} . $$ It arose when trying to perform a calculation on superconductivity. In particular I am interested in its ...
0
votes
1answer
114 views

Energy of repeated filter

For given sequences $a=(a_1, a_2, \cdots)$ and $b=(b_1, b_2, \cdots)$, define $$a \star b$$ as the convolution. Formally, $$c=a \star b$$ implies the $i$th element of $c$, $c_i$, satisfies the ...
40
votes
1answer
792 views

Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series $$f_{\sigma}(z) = ...
3
votes
2answers
121 views

series representation of bivariate functions

Given a bivariate function $f(x, y)$ with $x \in [-a,a]$ and $y \in [-b, b]$, what is the necessary and sufficient condition under which we can write $f(x, y) = \sum g_k(x)h_k(y)$ for all $(x,y)$ in ...
2
votes
2answers
345 views

Asymptotic behaviour of sequence

I am interested in the sequence $$a(n)=\sum_{k=0}^n {p(n-k) \choose k}$$ where $p(n)$ is a polynomial equation. When $p(n)=n$ this reduces to the Fibonacci sequence, but what about when $p(n)$ is ...
3
votes
1answer
329 views

Series of the inverse quadratic trinomial

Maybe it's a very simple question, but I have a problem with the following series $$\sum\limits_{n=1}^{\infty}\frac{1}{n^2+pn+q},$$ where $p, q \in \mathbb{R}$. I know about five ways how to calculate ...
3
votes
2answers
270 views

norm of the matrix series

The goal is to obtain an upper bound for the norm of the vector $$ \left\|\sum\limits_{k=0}^{\infty}(I−A)^kAw_k\right\| $$ for any symmetric matrix $A\in{\mathbb R}^{n×n}$ which $0\preceq A\preceq I$ ...
10
votes
1answer
800 views

Can an infinite number of mathematicians guess the number in a box with only one error?

In this question the following observation was made: Consider a sequence of boxes numbered 0, 1, ... each containing one real number. The real number cannot be seen unless the box is opened. Define ...
4
votes
1answer
177 views

Extending a Certain Result from Locally Convex Topological Vector Spaces to General Topological Vector Spaces

In this Math Stack Exchange post, I proved the following result. Theorem: Let $ X $ be a locally convex topological vector space. Let $ x \in X $ and suppose that $ (x_{n})_{n \in \mathbb{N}} $ is ...
1
vote
0answers
304 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}}} + ...
20
votes
3answers
1k views

Probabilities in a riddle involving axiom of choice

The question is about a modification of the following riddle (you can think about it before reading the answer if you like riddles, but that's not the point of my question): The Riddle: We assume ...