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Characterizing the integral as a function of $n$

Let $\alpha \in [0,3], \beta \geq 1, \lambda \geq 1$ and fix $n \in \mathbb{N}$. Consider the function $f(x;\alpha, \beta, \lambda) = x^{\alpha}\exp(-\lambda x^\beta)$. Let $I(n; \alpha,\beta,\lambda) ...
yfful's user avatar
  • 25
2 votes
2 answers
268 views

If $\inf\{b\in\mathbb{R}\mid\sum_{n=1}^{\infty}e^{-ax_n-by_n}<+\infty\}=1-a$ for all $a\in [0,1]$, does this equality hold for all $a\in\mathbb{R}$?

Let $\left\{x_n\right\}_{n=1}^{+\infty},\left\{y_n\right\}_{n=1}^{+\infty}\subset [0,+\infty)$ be two sequences of non-negative real numbers. Suppose there exist $\lambda\ge 1, c\ge 0$ such that $\...
YC Su's user avatar
  • 605
1 vote
2 answers
163 views

Transcendental functions with two prescribed values

Let $\alpha$ and $\beta$ two algebraic numbers lying in unit ball. Let $T:=(t_k)_k$ be an increasing sequence of positive integers such that $t_{k+1}/t_k$ tends to $1$ as $k\to \infty$. I would like ...
Jean's user avatar
  • 515
6 votes
3 answers
536 views

A need for analytic continuation of a finite sum function

Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$. I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum) \begin{align*} {\color{red}...
T. Amdeberhan's user avatar
3 votes
1 answer
166 views

A limit arising from Mellin Inversion: How to compute a specific term of an asymptotic series?

So I am wondering if there exists a general procedure for the following problem: given a monotonically increasing function $f(n)$ which is nonegative on the interval $[0,\infty)$ and grows faster than ...
Sidharth Ghoshal's user avatar
0 votes
0 answers
111 views

Sum power series not continuous unit circle

This is (probably) not a research question and I already asked it on StackExchange but I got no answer over there. Let us consider the sequence $(a_n)_{\geq 1} = \left(\frac{\cos(2\sqrt{2n}+\frac{\pi}{...
Libli's user avatar
  • 7,300
10 votes
2 answers
597 views

How to determine the asymptotics of $\sum_{n=0}^{\infty} e^{-\frac{2^n}{x}}$

I'm generally interested in being able to find an asymptotic expansion of $$ \sum_{n=0}^{\infty} \left[ e^{- \frac{f(n)}{x}} \right] $$ As $x \rightarrow \infty$ and $f(n)$ is a smooth monotonically ...
Sidharth Ghoshal's user avatar
1 vote
2 answers
113 views

$\log(t)$ term in the small time expansion of $\mathrm{Tr}( A e^{-tB} )$

Assume $A$ is an operator on a Hilbert space with discrete spectrum. Assume $B$ is a positive operator on the same Hilbert space also with a discrete spectrum. Also assume $A$ and $B$ commute. I'm ...
Fetchinson0234's user avatar
7 votes
1 answer
268 views

A differential equation governing compositional inversion

Looking for references for the following theorem. Given the formal Taylor series/exponential generating function $$T(z) = \sum_{n \ge 1} a_n \; \frac{z^n}{n!},$$ for which the indeterminates $a_n$ and ...
Tom Copeland's user avatar
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1 vote
1 answer
163 views

Upper bound on double series

We consider the sum $$ \sum_{m \in \mathbb Z^2} \frac{1}{(3 m_1^2+3m_2^2+3(m_1+m_1m_2+m_2)+1)^2}. $$ Numerically, it is not particularly hard to see that the value of this series is well below $4$, ...
Guido Li's user avatar
5 votes
1 answer
674 views

Is this infinite product entire?

Let $(z_i)$ be a square-summable sequence which is even summable but not absolute summable, i.e. $\sum_{i=1}^{\infty} \vert z_i \vert = \infty$,$\sum_{i=1}^{\infty} \vert z_i \vert^2 < \infty$ and $...
Guido Li's user avatar
4 votes
1 answer
317 views

Taylor coefficients of Hadamard product

I imagine this to be a very classical question in complex analysis: Consider the Hadamard product $$g(\mu) = \prod_{n=1}^{\infty}E_1(\mu z_n),$$ where $E_1(z):=(1-z)e^z$ is the first elementary ...
Guido Li's user avatar
1 vote
0 answers
76 views

Second question on a real sequence

I am thinking again about the real sequence $\{a_n\}_{n\ge1}$ which decays faster than any algebraic speed, that is, $\lim_{n\to \infty}n^pa_n = 0$ for every integer $p$. Actually, $a_n$ can be ...
Jacob Lu's user avatar
  • 903
4 votes
1 answer
692 views

An asymptotic expansion of a infinite sum

I am interested in the asymptotic expansion in $t$($t>0$) when $t\to 0^+$ of the following series $$ \sum_{k\ge 0}e^{-k^{2/n}t} $$ for integer $n>2$ (n=1 follows from Poisson summation formula ...
WhiteDwarf's user avatar
0 votes
1 answer
335 views

On partial sum estimate on the series $S(p,q;s)=\sum_1^q\frac{\sin^2(\frac{p\Gamma(n)}{n})}{n^s}$ and other Generalizations

Consider the following sum : $$S(p,q;s)=\sum_{n=1}^q\frac{\sin^2(\frac{p\Gamma(n)}{n})}{n^s}$$ Here , $p$ is a variable w.r.t which we are going to analyse the sum. $s$ is another parameter with ...
bambi's user avatar
  • 375
3 votes
1 answer
205 views

Understand the properties of this function

We define a function $f(t):=\sum_{n=0}^{\infty}e^{-nt}= \frac{1}{1-e^{-t}}= \frac{e^{\frac{t}{2}}}{e^{\frac{t}{2}}-e^{-\frac{t}{2}}}=\frac{2e^{\frac{t}{2}}}{\sinh\left(\frac{t}{2} \right)}$ observe ...
Kreimer's user avatar
  • 31
2 votes
1 answer
362 views

more on "sinc-ing" integrals and sums

This is a follow up on the MO question here. I kept being fascinated and bemused by these functions. Denote $\text{sinc}(x)=\frac{\sin x}x$. Experiments suggest that $$\sum_{n=1}^{\infty}\text{sinc}^...
T. Amdeberhan's user avatar
10 votes
2 answers
886 views

An attempt to generalize the previous inequality

In my previous MO question, the inequality was about a specific series and nicely answered by Cherng-tiao Perng. After testing with a few more numerical infinite sums, I came to realize that perhaps ...
T. Amdeberhan's user avatar
4 votes
1 answer
1k views

seeking proofs: infinite series inequalities

Question. Numerically, the following is convincing. However, is there a proof? $$\left(\sum_{k\geq1}\frac1{\sqrt{2^k+3^k}}\right)^4 <\pi^2\left(\sum_{k\geq1}\frac1{2^k+3^k}\right)\left(\sum_{k\...
T. Amdeberhan's user avatar
2 votes
0 answers
563 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 well-...
Stefan Kohl's user avatar
  • 19.6k