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0
votes
0answers
35 views

Explicit solution of recursive function for $F(n,a)=F(n-1,a)+F(n-(1+a),a), F(b,a)=1, b={1,2,3…,a+1}$ [on hold]

For the most common cases, $a=0$ and $a=1$, the explicit solutions are generally known as: $$F(n,0)=2^{n-1}$$ $$F(n,1)=\dfrac{\phi^n-(-\phi)^{-n}}{\sqrt{5}}$$ Is it possible to derive similar ...
6
votes
2answers
200 views

Finding a closed form for $\sum_{k=1}^{\infty}\frac{1}{(2k)^5-(2k)^3}$

I'm looking for a closed form for the expression $$ \sum_{k=1}^{\infty}\frac{1}{(2k)^5-(2k)^3} $$ I know that Ramanujan gave the following closed form for a similar expression $$ ...
4
votes
1answer
737 views

sum of the series $\sum_{n=1}^\infty \frac{(-1)^{(n-1)}}{\sqrt{n}}$?

Hi, We know that the series $\sum_{n=1}^\infty \frac{(-1)^{(n-1)}}{\sqrt{n}}$ is convergent and it is oscillating. and numerically it is almost 0.6048986434. I want to know what is the exact limit ...
-1
votes
0answers
72 views

Taylor-Series e^e^x [closed]

I'm new to the whole topic of Taylor-Series and I am trying to figure out the Taylor-Series of $e^{e^x}$. I got the derivatives but that doesn't help right now. I think I need the n-th derivative, ...
-2
votes
0answers
64 views

Could it be possible to check if Pi is a normal number? [closed]

So currently we don't know if Pi is a normal number and if it really contains all finite number sequences. Is it possible that we will know this in the future? Can we be sure one day?
1
vote
1answer
281 views

Rewriting a series $\sum_{n=0}^\infty \frac{1}{n!}(\Delta^\varepsilon)^n a_n$ in the form $\sum_{n=0}^\infty c_n \varepsilon^n$

I would like to rewrite the series $$\sum_{n=0}^\infty \frac{1}{n!}(\Delta^\varepsilon)^n a_n,$$ where $\Delta^\varepsilon=\sum_{k=1}^\infty \varepsilon^k b_k$, as a series in $\varepsilon$ ...
-5
votes
0answers
32 views

arithmitic and geometric sequences explicit recursive formula [closed]

Write an explicit formula for the following arithmetic sequence -4,-1,2,5... Write an recursive formula for the following arithmetic sequence -4,-1,2,5... Write an explicit formula for the following ...
1
vote
1answer
119 views

Extremal problem for sequences

Let $a_n$ be a sequence of positive numbers and define $$A_n=\sum_{k=1}^{n-1} a_k a_{n-k}.$$ I am interested in the supremum of the following quantity $X/Y$ where $$X=\sum _{i=1}^{\infty } \left(\sum ...
6
votes
1answer
625 views

How to find the coefficients of a poor-converging series?

I have the series $\psi(r,\theta;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n \theta/\Phi)$ and the boundary conditions $\psi(r,\pm\Phi;p)=\sum_{n=0}^{\infty} a_n J_{\pi ...
0
votes
0answers
63 views

How to prove that the convergence of $\sum_{n=1}^{\infty} \frac{\sec^a n}{n^c}$ implies that of $\sum_{n=1}^{\infty} \frac{\csc^a n}{n^c}$

The most general thing I've gotten is that the absolute convergence of $$\sum_{n=1}^{\infty} \frac{\csc^a (n + x)}{n^c}$$ implies that of $$\sum_{n=1}^{\infty} \frac{\csc^a \left(\frac{m}{2} n + ...
0
votes
0answers
48 views

a two dimensional integer bijection

I want to try to find a specific two-dimensional linear integer bijection. This is to be used in a double sum rearrangement. It's kind of complicated, but I would really appreciate if anybody has any ...
1
vote
0answers
44 views

Dirichlet series decomposition of arbitrary function

Originally asked on MSE here: http://math.stackexchange.com/q/1780149/52694 Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the original ...
1
vote
1answer
245 views

Is this function $\mathbb{Q}$ periodic?

Considering a function $f$ exponentially decreasing at infinity, is the following function $\mathbb{Q}$ periodic ? $$F(x)= \sum\limits_{q =1}^{\infty} \; \sum\limits_{n =1}^{\infty} \; ...
3
votes
1answer
168 views

On construction of a $\mathbb{Q}$ periodic function with Fourier series

Taking $f$ a function decreasing exponentially at infinity we can consider the periodic function given by following Fourier series: $$F(x)= \sum\limits_{n =1}^{\infty} f(n) e^{2 i \pi n x}$$ Using ...
3
votes
0answers
177 views

First to note/document the relation between permutohedra and multiplicative inversion

The relation between the refined face numbers of the permutohedra and the formal series expansion of the reciprocal of a function (exponential generating function, formal Taylor series) is given in ...
2
votes
0answers
66 views

Relating face polytopes of permutohedra to integer partitions

The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an ...
6
votes
2answers
484 views

How many integers are of the form $n/d(n)$, where $d(n)$ is the number of divisors of $n$?

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 ...
1
vote
0answers
45 views

Computing Moore-Penrose generalized inverse using convergent geometric series

I came across an article (published in IEEE transactions or Wiley publications) which states that moore-penrose generalized inverse can be computed using the following two equations. Let $A$ be an ...
3
votes
1answer
130 views

Generalized Equal Distribution Kolakoski Sequence Conjecture

If we let $\operatorname{Kol}(a_1,\dots,a_n)$ be the run sequence determined by the rules of Kolakoski Frequencies, we ask is there a sequence of $\operatorname{Kol}$ that DOES NOT obey the $1/n$ ...
-1
votes
1answer
167 views

For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$

Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
1
vote
1answer
61 views

Inequality implies locally uniform convergence of a series

We have the inequality $$\alpha_n(t) \le 4\pi(3\pi)^{1/3} \exp\left\{\int_0^t(1+3F_P(\sigma)) \, d\sigma \right\} \cdot \int_0^t P(\sigma)^2(3CD_1^2)^{1/3}\alpha_{n-1}(\sigma) \, d\sigma$$ for ...
7
votes
2answers
237 views

How to estimate a specific infinite matrix sum

Let $M$ be an $n$ by $n$ matrix with each diagonal element equal to $k$ and each non-diagonal element equal to $k-1$ where $n$ and $k$ are positive integers. Let $k < n$ and we can assume both $k$ ...
2
votes
0answers
77 views

How to estimate $\prod_{t=1}^{N}\frac{1}{2-z^t}$ for large $N$?

Based on the top answer to How to estimate of $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for large $N$? Can anyone find an approximate closed form for $$ ...
4
votes
2answers
289 views

Is there a closed form for $\sum_{k=0}^n \frac{x^k}{k!}$? [closed]

What is the closed form of $$\sum_{k=0}^n \frac{x^k}{k!}$$ as a function of $x$ and $n$? Knowing that it converges to $e^x$ when $n\to \infty$.
4
votes
2answers
229 views

Moment problem on [-1,1]: necessary and sufficient conditions

Consider a sequence of real numbers $s=(s_0,s_1,\ldots)$. When is there a Borel measure $\mu$ supported on $[-1,1]$ so that $$ s_k = \int_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;? $$ ...
8
votes
0answers
133 views

Summation of series involving $\sinh$ of a square root

Consider the following series: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + m^2}/2)}{\sinh( \pi \sqrt{n^2 + m^2})} $$ From the physical ...
-4
votes
1answer
137 views

Does $\sum_n \frac{\sin n}n$ converge absolutely? [closed]

Using Dirichlet's test, one can prove that $\sum_{n\geq 1} \frac{\sin n}n$ converges. Does it converge absolutely?
4
votes
2answers
2k views

Ratio of Sequences Sum Inequality

I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i ...
6
votes
4answers
424 views

The coefficient of a specific monomial of the following polynomial

Let the real polynomial $$f_{a,b,c}(x_1,x_2,x_3)=(x_1-x_2)^{2a+1}(x_2-x_3)^{2b+1}(x_3-x_1)^{2c+1},$$ where $a,b,c$ are nonnegative integers. Let $m_{a,b,c}$ be the coefficient of the monomial ...
2
votes
1answer
97 views

Sufficient conditions for weak majorization

Suppose $x_1\ge x_2\ge \cdots \ge x_n\ge 0$ and $y_1\ge y_2\ge\cdots\ge y_n\ge0$ be reals such that for any positive integer $p$, $$ \sum_{i=1}^n x_i^p \geq \sum_{i=1}^n y_i^p. $$ Question: Is ...
9
votes
1answer
231 views

How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently

Let $M$ be a real symmetric integer valued positive definite matrix with $\det(M) \geq 1$. I would like write code to compute $$S_M= \sum_{x \in \mathbb{Z}^n} e^{-x^TMx}.$$ One option is to simply ...
4
votes
1answer
179 views

Non-equivalence of admitting different types of bases in Banach spaces

Whenever a certain type of (Schauder) basis is defined, it is natural to ask where that type lies in the scheme of other types of bases. This involves finding counter-examples of one type of basis ...
1
vote
2answers
156 views

aproximate sum involving binomial coefficients

I have the problem for computing the j-derivative of a logarithm, with $j\gg1$ \begin{equation} c_j=\left.\frac{\partial^j}{\partial s^j}\log\left(1+Ae^s+Be^{2s}\right)\right|_{s=0}, \end{equation} ...
15
votes
5answers
613 views

Asymptotics of a recurrence relation

The sequence $(a_n)_{n \ge 0}$ satisfies, $a_0 = a_1 = 1$ and the recursion relation: $$a_n = \sum\limits_{k=0}^{[n/2]} \frac{a_k}{(n-2k)!}$$ where, $[x]$ is the nearest integer to $x$ not exceeding ...
2
votes
0answers
101 views

Enumerating the number of degree d curves tangent to a planar conic

This question is based on a special case of the Coparaso Harris formula, as described in Counting curves on rational surfaces - R. Vakil. Let $E$ be a non-singular planar conic. Then every degree ...
4
votes
2answers
813 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}} + ...
37
votes
4answers
2k views

Why do Pell equations appear in Ramanujan's pi formulas?

While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula. I. Given the fundamental unit ...
6
votes
2answers
914 views

Divergence of Dirichlet series

Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge? I asked ...
5
votes
1answer
342 views

An elementary inequality for a recursive double sequence

Here is what looks like (but is not) an Olympiad problem. Is it really that tough, or am I overlooking a simple solution? I have a system of sequences $\sigma_0(m),\sigma_1(m),\ldots\,$ defined for ...
34
votes
5answers
4k views

Is there an “elegant” non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?

Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway... Consider this problem. I've been trying to find a formula to expand the "regular iteration" of ...
15
votes
3answers
372 views

Evaluating an infinite sum related to $\sinh$

How can we show the following equation $$\sum_{n\text{ odd}}\frac1{n\sinh(n\pi)}=\frac{\mathrm{ln}2}8\;?$$ I found it in a physics book(David J. Griffiths,'Introduction to electrodynamics',in ...
3
votes
0answers
159 views

Combination of Generating Functions

Suppose I have the following generating functions: $$\frac{x^ke^{\left(z-\frac{1}{N}\right)x}}{N^{k-1}k!\sum_{j=0}^{N-1}w_N^{-jk}e^{\frac{w_N^jx}{N}}}=\sum_{j=0}^\infty H_{N,k,j}(z)\frac{x^j}{j!}$$ ...
3
votes
1answer
88 views

How to show monotonocity and the limit? [closed]

Let me reformulate my recent question. Let $n, N$ denote density and cdf of Gaussian distribution. Let us consider its modification, given by density: $$\phi(x) = C\left\{ \begin{array}{lcc} ...
25
votes
7answers
3k views

What problem would you base your mathcoin on?

Recently, a variant of electronic currency, based on prime sextuplets, broke the record in generating the largest known set of six primes, packed as closely as possible, that is, a sextuple ...
1
vote
0answers
44 views

Bounds on $\sum_{j=1}^m\frac{\pi^j}{\Gamma(j)(x^2+(j+1/4)^2)}$

During our search of real rooted entire function approximations to Riemann $\Xi$ function, we need to calculate the upper and lower bounds of $$f_m(x):=\sum_{j=1}^m ...
4
votes
1answer
246 views

Can we always attain another prime via inserting digits between the digits of a fixed prime?

The sequence OEIS A080437 is For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m. I don't see why ...
1
vote
0answers
126 views

Generating a series representation for the inverse of the operator $f(f)$

I am considering the following problem: Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...
2
votes
2answers
82 views

Sum of subsequence, over index set of non-zero density, of monotone divergent sum also divergent?

For a subset $S$ of the natural numbers $N$ and $n\in N$ let $|S\cap n|$ be the number of members of $S$ that are less than $n$. Suppose $S$ does not have upper asymptotic density $0$. That is, ...
3
votes
1answer
274 views

Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that ...
9
votes
1answer
442 views

Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?

It is a result of folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. Šalát's paper: ...