All Questions
12
questions
7
votes
0
answers
189
views
Combinatorial interpretation of rational function on e
Over the last few weeks I have become obsessed with expressions like
$$
\frac{e+4 e^{2}+e^{3}}{(1-e)^{4}},
$$
$$
\frac{e+26 e^{2}+66 e^{3}+26 e^{4}+e^{5}}{(1-e)^{6}},
$$
or
$$
\frac{e+120 e^{2}+1191 e^...
3
votes
1
answer
406
views
Computing the Hardy-Ramanujan asymptotic formula using method of steepest descent/saddle point method
I am trying to obtain and prove the Hardy-Ramanujan asymptotic approximation formula given by:
$$p(n) \sim \frac{1}{4n\sqrt{3}}e^{\pi\sqrt{\frac{2n}{3}}},$$
by using Dedekind's eta function
$$\eta(z)=...
1
vote
1
answer
399
views
Will there be a closed form expression for all $\zeta(2n+1)$?
It is known that $\zeta(2)=\frac{\pi^{2}}{6}$ and that $\zeta(4)=\frac{\pi^{4}}{90}$. Thus, for $\zeta(2n)$, this can be generalized to :
$$
\zeta(2n)=\frac{(-1)^{n-1}B_{2n}(2\pi)^{2n}}{2(2n)!}
$$
...
user815114
3
votes
1
answer
81
views
In how many ways can I express a positive integer as a sum of elements in a subset of $\mathbb Z^+$?
Let $S\subseteq \mathbb Z^+$ be set of positive integers. Given $n\in\mathbb Z^+$, how can I find the number of ways in which we can express $n$ as a sum of elements in $S$? ($S$ can be infinite.)
$$
...
- 7,534
3
votes
1
answer
329
views
Weierstrass elliptic funcion in Laurent series form
Could anyone please help me to figure out how
$$f_0(z) = \wp(\log z; i\pi, \log \rho)$$
where $\wp$ denotes the Weierstrass elliptic function and $i \pi$ and $\log \rho$ are its half periods.
$$ f_0(...
- 33
0
votes
1
answer
33
views
How to show that $\frac{4^n}{n^{3/2}\sqrt \pi}$ could not be expressed as $\sum_i^m p_i(n)\lambda_i^n$
How to show that
$$
\frac{4^n}{n^{3/2}\sqrt \pi}
$$ has not the form $p_1(n) \lambda_1^n + \ldots p_i(n) \lambda_i^{n}$ for some polynomials $p_i(n)$ and numbers $\lambda_i$?
- 18.2k
4
votes
1
answer
569
views
Relation between Ramanujan Theta Function and Jacobi Theta Function
In the theory of $q-$series,
we have Ramanujan Theta function
\begin{align}\label{rama-theta}
f(a,b):=\sum_{n=-\infty}^{\infty}
a^{\frac{n(n+1)}{2}}b^{\frac{n(n-1)}{2}}
,\qquad ...
- 641
3
votes
0
answers
179
views
Transformation of $e^{2\pi i z}$.
I want to find the use of $q=e^{2\pi i z}$ in modular forms and combinatorics.
The question here maybe a little not specific,
and I'll explain it.
I know that this is a exponential map，and map upper ...
- 641
2
votes
0
answers
88
views
Sum of Roots of Unity With Weighted Exponents
I have the following conjecture that I want to believe has some sort of classical result associated to it, but have yet to find any such evidence.
Let $\ell,r\in\mathbb{Z}^+$, and fix $w_1,\ldots,w_{\...
- 31
8
votes
1
answer
990
views
Is there an upper bound on the growth rate of analytic functions?
This problem comes from a solution tactic used in Is there a rational surjection $\Bbb N\to\Bbb Q$?, where I discovered that there is an analytic function $f(z)$ that takes the values $f(n)=a_n$ for ...
- 27.7k
12
votes
3
answers
3k
views
How to prove Euler's pentagonal theorem? Some hints will help
Euler's pentagonal theorem is the following equation:
$\prod\limits_{n=1}^{+\infty}(1-q^n)=\sum\limits_{m=-\infty}^{+\infty}(-1)^m q^{\frac{3m^2-m}{2}}$
where $|q|<1$ is a complex number.
I ...
- 2,900
1
vote
1
answer
361
views
Representability as a Sum of Three Positive Squares or Non-negative Triangular Numbers
Let $r_{2,3}(n)$ and $r_{t,3}(n)$ denote the number of ways to write $n$ as a sum of three positive squares (A063691) and as a sum of three non-negative triangular numbers (A008443), respectively. I ...
- 17.1k