I am trying to solve the recurrence relation $a_{n}=\alpha a_{n-1}^2+\beta a_{n-1}$ where $\alpha$ and $\beta$ are constants. I have been trying to find specific techniques for solving this equation in closed form. I have not come across any such work, yet. I may not be looking into the proper literature as I guess this problem to be a well studied one. Any pointer towards the solution and/or the direct solution will be really helpful.
Thanks.
Your recurrence equation is called a Quadratic map,most quadratic maps don't have a closed form solution.