Let $n\ge 2r$ be positive integers.
Is there a closed form for following finite summation involving in qbinomial coefficients
$$\sum_{s=0}^r(1)^sq^{\frac{s(s+1)}{2}}{n2r+s\brack n2r}_q{n\brack rs}_q\,\,\,\,\,\, ?$$
I found this while studying qFibonacci/ Lucas polynomials.
What is the general approach for evaluating this type of series?
Any suggestion would be appreciated.

$\begingroup$ What is your definition of $\big[..\big]_q$ ? $\endgroup$ – Konstantinos Kanakoglou Jan 23 '17 at 1:10

$\begingroup$ I use the definition $[n]_q=1+q+q^2+\cdots+q^{n1}$ for qintegers . $\endgroup$ – Bumblebee Jan 23 '17 at 1:15

1$\begingroup$ Out of curiosity: where did you find this sum? I mean did you find it mentioned in some paper or is it a byproduct of some calculations of yours? It would be helpful to add some more context. $\endgroup$ – Konstantinos Kanakoglou Jan 23 '17 at 2:04

$\begingroup$ I an also very curious about the sum  it looks like the major index statistic on some family of lattice paths with some restriction (using inclusionexclusion)... $\endgroup$ – Per Alexandersson Jan 26 '19 at 22:33
Doron Zeilberger has written a Maple code for checking and proving ordinary binomial identities and their $q$analogues. What you need in the present case is the package called qEKHAD.
I just tested your sum and it leads to a quadratic recurrence in $r$, so you may not expect a closed form as an answer. The lucky cases are linear recurrences in the "external" parameter (here, it is $r$). The latter situation fits the socalled WilfZeilberger pair.
I also sought for other factors $q^{\mu(s)}$ instead of $q^{\binom{s+1}2}$, none of the usual suspects lead to a closed form. Sorry.

$\begingroup$ Thank you for your help. I am tring to simplify this summation bit further. Numerical values suggest that above sum equal to $\sum_{s=0}^{r1}(1)^{s+1}q^{\frac{s(s+1)}{2}}{n2r+s\brack n2r}_q{n\brack rs1}_q.$ And also I fund that this equality holds for ordinary binomial coefficients (in which $q=1$). Do you think that they are actually equals for qbinomials? $\endgroup$ – Bumblebee Jan 23 '17 at 3:10

2$\begingroup$ I just checked: the two sums satisfy the same quadratic recurrence (in $n$) and matching initial conditions. They are indeed equal. $\endgroup$ – T. Amdeberhan Jan 23 '17 at 3:56

$\begingroup$ I found this paper by Wilf and Zeilberger. Does the same algorithm works for qanalogies? $\endgroup$ – Bumblebee Jan 24 '17 at 1:32
