All Questions
6 questions
7
votes
0
answers
182
views
Positivity of certain polynomial coefficients
Consider the rational functions (in fact, polynomials)
$$F_n(q)=\frac1{(1-q)^{2n}}\sum_{k=0}^n(-q)^k\frac{2k+1}{n+k+1}\binom{2n}{n-k}
\prod_{j=0,\,j\neq k}^n\frac{1+q^{2j+1}}{1+q}.$$
The numbers $\...
- 43.2k
5
votes
2
answers
293
views
Positivity of coefficients of a polynomial derived from Schubert polynomials
Let $W=\bigcup_{n=1}^\infty S_n$ be the union of all symmetric groups $S_n$. For an element $w\in W$, denote by $\mathfrak{S}_w$ the Schubert polynomial associated to $w$, and by $\partial_w$ the ...
- 1,066
4
votes
1
answer
239
views
A discrete operator begets even/odd polynomials
Given a function $f(x)$ define the forward shift operator by $Ef(x)=f(x+1)$ and the discrete derivative $\delta f(x)=(E-1)f(x)=f(x+1)-f(x)$.
Given a partition $\lambda=(\lambda_1,\lambda_2,\dots,\...
- 43.2k
6
votes
0
answers
115
views
Recursions which define polynomials?
Let $k$ be a positive integer and let
$$h(n,k,q)=\frac{1-(1+q^{k})q^{2k(n-1)+1}+q^{2}}{1-q^{2n-1}}h(n-1,k,q)-\frac{(1-q^{k(2n-3)})(1-q^{2k(n-1)})q^2}{(1-q^{2n-1})(1-q^{2n-3})}h(n-2,k,q)$$
with ...
- 5,622
11
votes
0
answers
361
views
Positivity of polynomial sequences via generating series
In this question I address
the problem of proving the nonnegativity of a numerical sequence
$a_0,a_1,a_2,\dots$ via generating series technique. In the notation
$A(x)=\sum_{n=0}^\infty a_nx^n\ge0$ ...
- 13.5k
16
votes
0
answers
910
views
Polynomials with presumably positive coefficients
After seeing that some positivity problems get their solutions on MO,
I am quite enthusiastic of posing my (and not only) problem of positive flavour.
In order to state it, I have to introduce the ...
- 13.5k