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Tagged with determinants q-analogs
3 questions with no upvoted or accepted answers
6
votes
0
answers
203
views
Conjecture for a certain Cauchy-type determinant
Given the Cauchy-like matrix
$$
\mathbf X_M(q) = \left[ \frac{2}{\pi} \frac{
\Gamma\!\left(m - \frac{1}{2} \right)\Gamma\!\left(n + \frac{1}{2} \right)
}{
\Gamma(m)\,\Gamma(n)
}
\frac{m-\frac{3}{4}}
{\...
- 3,756
4
votes
0
answers
113
views
Positivity of q-analogs of central binomial coefficients?
With the usual $q-$notations
$[n]_q=1+q+\cdots+q^{n-1}=\frac{\,\,1-q^n}{1-q},$
$[n]_q!=[1]_q[2]_q\cdots[n]_q$ and
$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}$
let
$$b(n,k,r,q)=\det\left(q^{r\...
- 5,622
1
vote
0
answers
88
views
Evaluate $\det[[\lfloor\frac{aj-(a+1)k}n\rfloor]_q]_{1\le j,k\le n}$ and $\det[[\lceil\frac{(a+1)j-ak}n\rceil]_q]_{1\le j,k\le n}$
The $q$-analogue of an integer $m$ is defined by
$[m]_q=(1-q^m)/(1-q)$. Note that $\lim_{q\to1}[m]_q=m$.
I have formulated the following conjecture on determinants involving the floor function and the ...
- 15.6k