I have a sequence of polynomials $\{ f_{m} \}$ with $f_{m} \in \mathbb{Z}[x]$ that satisfies an order 2 linear recurrence relation. In particular, I have a polynomial $g \in \mathbb{Z}[x]$ such that $$ f_{m+1} - g \cdot f_{m} + f_{m-1} = 0$$
subject to some initial conditions. A little linear algebra gives a closed form function $f \colon \mathbb{R}^+ \times \mathbb{C} \to \mathbb{C}$ such that $f(m, x) = f_m(x)$ whenever $m \in \mathbb{N}$. The choice of domain will be clear in a second.
Questions:
- Some experimentation with computer algebra leads me to believe that each $f_m$ is irreducible over $\mathbb{Z}$. In general, is there any way to relate the function $f$ to the irreducibility of the $f_m$'s? Ie, are there any theorems that say "If $f$ has property [blah], then $f_m$ is irreducible"?
- Is there any sort of nice relationship between the zeroes of each $f_m$ and the set $f^{-1}( 0 )$? I think that $f^{-1}(0)$ is a collection of embedded paths in $\mathbb{R}^+ \times \mathbb{C}$. The degree of $f_{m+1}$ is larger than the degree of $f_m$, so these paths must either branch somewhere between $m$ and $m+1$ or "new arcs" must appear (thinking about the $m$-coordinate increasing continuously). It would be particularly nice if there was a path in $f^{-1}(0)$ from a root of $f_{m+1}$ [a point $(m+1, x) \in f^{-1}(0)$] to a root of $f_{m}$, but for some reason, this seems unlikely.
In general, any references or information about the relationship between $f$ and the polynomials $f_m$ would be greatly appreciated.