I have bumped into the following expressions involving $q$-binomial coefficients.
$$
\sum\_{s=0}^a (-1)^s q^{s^2-s} \left(\begin{array}{c}2b+1-2s\\2a-2s\end{array}\right)\_q
 \left(\begin{array}{c}b\\s\end{array}\right)\_{q^2}
$$
The expressions depend on $a$ and $b$ and are zero unless $0\leq a\leq b$. 


Have these expressions appeared before? What is known about them? Any references would be appreciated.



Expressions involving $q$-binomial coefficients?
This is a summable ${}\_2\phi\_1$. The standard reference is Gasper and Rahman, Basic Hypergeometric Series. It looks like your series is
$$\frac{(q;q)\_{2b+1}}{(q;q)\_{2a}(q;q)\_{2b+1-2a}}{}\_2\phi\_1\left(\begin{matrix}q^{-2a},q^{1-2a}\\q^{-2b-1}\end{matrix};q^2,q^{4a-2b-2}\right).$$
By the $q$-Gauss summation, this factors completely as
$$\frac{(q;q)\_{2b+1}}{(q;q)\_{2a}(q;q)\_{2b+1-2a}}\frac{(q^{-2b-2+2a};q^2)\_a}{(q^{-2b-1};q^2)\_a}.$$
If you compute your expression for some special values of $a$ and $b$ you should see this factorization; otherwise I made a mistake.


