# 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 $$b(n,k,r,q)=\det\left(q^{r\binom{i-j}2}\frac{[2i+k+1]_q}{[i+j+k]_q}\binom{i+j+k}{i-j+1}_q\right)_{i,j=0}^{n-1}.$$

It can be shown that $$b(n,k,1,q)=\binom{2n+k-1}{n}_q.$$ Therefore $$b(n,k,1,q)$$ has positive coefficients as a polynomial in $$q$$ for each positive integer $$k.$$

Computations suggest that also $$b(n,k,0,q)$$ and $$b(n,k,2,q)=q^{n(n+k-1)}b(n,k,0,1/q)$$ have positive coefficients.

Any idea how to prove this?