Question

Studying a specific quantum cluster algebra, I have come across the following sequence of polynomials :

$$X_1$$ $$q^{-1/2}(X_1X_2-1)$$ $$q^{-1/2}(X_1X_2X_3-X_3-X_1)$$ $$q^{-1}(X_1X_2X_3X_4-X_3X_4-X_1X_2-X_1X_4+1)$$ $$q^{-1}(X_1X_2X_3X_4X_5 - X_1X_4X_5 - X_1X_2X_3 - X_1X_2X_5 - X_3X_4X_5 + X_1 + X_3 + X_5)$$

Either finding a simple expression, or just other places where these polynomials occur would be interesting. They differ only from a $q$ factor from their commutative analogues, which can be caracterized as determinants of the following matrix:

$$\begin{pmatrix} x_n & 1 & 0 & \cdots & 0 \\ 1 & x_{n-1} & 1 & \cdots & 0\\ 0 & 1 & x_{n-2} & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & x_1 \end{pmatrix}$$

I read about a quantum analog of determinants, but it involves quantum matrices and this is not a quantum matrix since it would need to satisfy $X_1 = qX_1$.

Answer 1

$$q^{-[n/2]/2}\sum_{k=0}^{[n/2]}(-1)^k \sum X_{j_1}\dots X_{j_{n-2k}},$$ where the second sum is over sequences $0=j_0,j_1,\dots ,j_ {n-2k},j_{n-2k+1}=n+1$ such that $j_0< j_1< \dots< j_ {n-2k}< j_{n-2k+1}$ and $j_{l+1}-j_l$ is odd for $l=0,\dots,n-2k$. Is not this a simple expression you are searching for?

Answer 2

They look a bit like continuants, q.v., which come up in continued fractions. See, e.g., http://en.wikipedia.org/wiki/Continuant_(mathematics)