Oscillation of a family of qq-difference equations

We obtain the complete classification of oscillation and nonoscillation for the qq-difference equation xΔΔ(t)+b(−1)ntcx(qt)=0,b≠0, where t=qn∈T=qN0,q>1t=qn∈T=qN0,q>1, c,b∈Rc,b∈R. In particular we prove that this qq-difference equation is nonoscillatory, if c>2c>2 and is oscillatory, if c<2c<2. In the critical case c=2c=2 we show that it is oscillatory, if |b|>1q(q−1), and is nonoscillatory, if |b|≤1q(q−1).