I have a question about common divisors of some expressions involving Gaussian coefficients, in particular in the case ${n \brack 1}_{q} = \frac{q^{n}-1}{q-1}$ where $q$ is a prime power.
It is well known that the greatest common divisor $$\gcd{\left( {a \brack 1}_{q}, {b \brack 1}_{q}\right)} = {d \brack 1}_{q}$$ where $d = \gcd{(a,b)}$.
If we assume that $a,b \geq 1$, $a+b < n$, is there an easy way to determine the following? $$\gcd{\left({n \brack 1}_{q},\ q^{a}{b \brack 1}_{q} \pm 1\right)}$$
This has come up in some research I am doing. In particular I am hoping to find a nice way to determine when this $\gcd$ is 1.
I do know that if $n = a+kb+s$ for some $0 \leq s < b$ then $${n \brack 1}_{q} = q^{a}{b \brack 1}_{q}q^{s}\frac{q^{kb}-1}{q^{b}-1} + {s \brack 1}_{q}$$ which I'm hoping can help.