I've been working on a problem in economics that involves finding the roots of the polynomial
$$ P(i) = \sum_{n=0}^t \bigg[ i^{n+1} \cdot \bigg(\binom {t}{n+1} + \alpha \binom {t}{n} \bigg ) \bigg ]$$
where $t \in \mathbb{N}$, $\ \alpha \in \mathbb{R} < 0$, and $i > 0.\ $ I want to be able to take $t$ and $\alpha$ from the problem and quickly approximate $i$. Since the polynomial has a factor of $i$, $i=0$ is always a solution, but for the application, $i$ must be positive.
I know that there is some relation between adding two choose functions together, but I am worried that the factor of $\alpha$ would break the relationship. Can the expression for the coefficients be simplified at all?
$$ {t \choose n+1} + \alpha {t \choose n} = \left(\frac{t-n}{n+1}+\alpha\right) {t \choose n} $$