Rational difference equation

Remarks

Example: $x_{n+1} = 1/(1 + x_n)$ is a rational difference equation with $k = 0$, $\alpha = 1$, $\beta_0 = 0$, $A = 1$, $B_0 = 1$. For $x_0 = 1$ all denominators are at least $1$, and $x_n = F_{n+1}/F_{n+2}$ with $F$ the Fibonacci numbers. Non-example: $x_{n+1} = 1/(1 + x_n^2)$ is not of this form, since its denominator is not affine in $x_n$. If all $B_i = 0$ and $A \ne 0$, the equation is linear; the interesting cases have some $B_i \ne 0$. The case $k = 0$, $x_{n+1} = (a x_n + b)/(c x_n + d)$ with $c \ne 0$ and $ad - bc \ne 0$, is the Riccati difference equation.