All Questions
8 questions
3
votes
1
answer
495
views
Simple closed forms for sums such as $\sum_{k=1}^\infty \frac{(-1)^{k+1}}{qk - p}$ and related integrals
My goal here is to get a simple expression for $\zeta(3)$. This is a follow up to my previous question posted here. Any Taylor-like expansion from everything I tried won't make it. So this is my last ...
- 3,098
1
vote
1
answer
151
views
Original examples of functions of slow increase in the spirit of Jakimczuk
I believe that it is possible to prove that $$f(x)=e^{\operatorname{Ai}(x)}\log x$$ is a function of slow increase in the spirit of the definition given by the author of [1], where $\operatorname{Ai}(...
1
vote
0
answers
69
views
Recurrence involving families of orthogonal polynomials
Let $ \forall n \in N, n\geq 1$ $$ R_n(x)=(-1)^n n! \displaystyle \frac{(x-1)...(x-n)}{(x(x+1)..(x+n))^2}$$ thus by decomposition in simple element it's easy to see that
$$ (1): \quad R_n(x)= \...
- 417
6
votes
1
answer
363
views
Double Series involving Gamma function
Does anyone have any ideas on howto verify $$\sum_{n,m=0}^\infty \frac{\Gamma(n+m+3x)}{\Gamma(n+1+x)\Gamma(m+1+x)}\cdot \frac{1}{3^{n+m+3x-1}} = \Gamma(x)$$ for $x>0$?
I posted this question also ...
- 284
3
votes
1
answer
205
views
Understand the properties of this function
We define a function
$f(t):=\sum_{n=0}^{\infty}e^{-nt}= \frac{1}{1-e^{-t}}= \frac{e^{\frac{t}{2}}}{e^{\frac{t}{2}}-e^{-\frac{t}{2}}}=\frac{2e^{\frac{t}{2}}}{\sinh\left(\frac{t}{2} \right)}$
observe ...
- 31
3
votes
1
answer
105
views
How to show monotonocity and the limit? [closed]
Let me reformulate my recent question.
Let $n, N$ denote density and cdf of Gaussian distribution. Let us consider its modification, given by density:
$$\phi(x) = C\left\{ \begin{array}{lcc}
\sqrt{...
- 31
7
votes
1
answer
310
views
Asymptotic behavior of a sequence of functions
For $n\in\mathbb{N}$ and $q\in(0,1)$, define
$$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$
...
- 2,188
11
votes
1
answer
1k
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 $a=1/4$.
...
- 1,649