All Questions
8 questions
27
votes
3
answers
2k
views
Kasteleyn's formula for domino tilings generalized?
It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$.
Kasteleyn's ...
- 43.1k
21
votes
2
answers
2k
views
Real rootedness of a polynomial
Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$, and the polynomial given by:
$$ P_{m,n}(t) := \sum_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$
I've found with Sage that for every $...
- 1,889
4
votes
1
answer
223
views
Asymptotics for 'generalized" Kasteleyn's formula
A follow up on an earlier MO question.
Kasteleyn's formula for the number of domino tilings of a $2n\times 2n$ square
$\prod_{j=1}^n\prod_{k=1}^n \left( 4\cos^2(\pi j/(2n+1))+4\cos^2(\pi k/(2n+1))\...
- 43.1k
2
votes
2
answers
269
views
Ratios of polynomials and derivatives under a certain functional
Let $p(x)$ be a polynomial of degree $n>2$, with roots $x_1,x_2,\dots,x_n$ (including multiplicities). Let $m$ be a positive even integer. Define the following mapping
$$V_m(p)=\sum_{1\leq i<j\...
- 43.1k
2
votes
0
answers
231
views
Where does this trig. identity hold?
Fix an integer $n\geq1$.
QUESTION. Is it possible to find ALL pair of sequences of non-negative integers $(a_k,b_k)$, for $k=1,2,\dots,n$, such that
$$\sum_{k=1}^n \sin^{2a_k}\theta\cdot \cos^{2b_k}\...
- 43.1k
1
vote
0
answers
134
views
Number of solutions to a diophantine equation
Given a positive integer $n$, consider the diophantine equation $4x^2+y^2+4x+y=2n$ with solutions in non-negative integers $x$ and $y$.
Define the proportion
$$\delta_n=\frac{\#\{(x,y)\in\mathbb{Z}^2_{...
- 43.1k
0
votes
1
answer
129
views
Seeking an integral formulation for an algebraic function
While working with a generating function for the Catalan numbers, I came across the integral representation
$$\frac1{1+\sqrt{1-4x}}=\frac1{2\pi}\int_0^{\infty}\frac{\sqrt{t}}{(t+\frac14)(t-x+\frac14)}\...
- 43.1k
0
votes
0
answers
161
views
Asymptotic analysis of a sum of complex summands using integral
I'm trying to find the exact asymptotics of a sum:
$$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$
as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, |y|\...
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