I'm reading this paper and have a question about the math done on page 4.
We go from having
$$\lambda^{T_0} = p \lambda^{T_{0 + 1}} + q\lambda^{T_{0 - 1}}$$ to
$$p \lambda^2 - \lambda + q = 0$$
How does this reduction work? It's unclear to me how we lose the distinguishers between $\lambda^{T_0}, \lambda^{T_{0 + 1}}$, etc. I tried rearranging so that I had
$$0 = p \lambda^{T_{0 + 1}} - \lambda^{T_0} + q\lambda^{T_{0 - 1}}$$
but wasn't sure what I should do after this. Any help is appreciated.
Also, where can I read about how the power function becomes $q_{T_0} = \lambda^{T_0}$? I know what a power function is and understand finding a constant ratio, but I've never seen one put in terms of a $\lambda$. Thanks in advance!
What it says is
$\lambda^{T_0} = p \lambda^{T_{0 }+ 1}+ q\lambda^{T_{0 }- 1} $.
Now you can divide by $\lambda^{T_{0 }- 1} $ to get that quadratic.
To make it clearer, I'll replace $T_0$ by $n$.
What it says is
$\lambda^{n} = p \lambda^{n+ 1}+ q\lambda^{n- 1} $.
Now you can divide by $\lambda^{n- 1} $ to get that quadratic.