Chebyshev polynomials have real roots and satisfy a recurrence relation. I was wondering if one can find a sequence of polynomials with integral or rational roots with similar properties. More precisely, one is looking for a sequence of polynomials $(f\_n),f\_n\in\mathbf{Q}[t]$ such that


1. $\deg f\_n\to\infty$ as $n\to\infty$;
2. $\sum\_{n=0}^\infty f\_n(t) x^n$ is (the Taylor series of) a rational function $F$ in $x$ and $t$.
3. All roots of any $f\_n$ are integer and have multiplicity 1. (A weaker version: the roots are allowed to be rational and are allowed to have multiplicity $>1$ but there should be an $a>0$ such that the number of distinct roots of $f\_n$ is at least $a\deg f\_n$.)



Chebyshev-like polynomials with integral roots
Here's one thought. For each integer k, f\_n(k) satisfies a recurrence relation. If the roots of f\_n are all integers, then f\_n(k) and f\_n(k+1) have to be "in sync" in the sense that they never have opposite sign. This is a strong condition! For instance, suppose the sequences f\_n(k) and f\_n(k+1) each have unique largest eigenvalue: then these eigenvalues would have to have the same argument.


**Update:** Qiaochu's answer suggests that in fact working mod p would be just better than the "archimedean" picture sketched above, since it is really F\_q[t], not Z[t] or Q[t], that is analogous to the integers. Let F\_n(t) be the reduction of f\_n(t) to F\_p[t]. If f\_n(t) has all roots rational for every n, then the reduction of f\_n(t) mod p has the same property. But now we are saying something quite strong; that f\_n(t) lies in a finitely generated subgroup of F\_q(t)^\*! This is presumably ruled out by Mason's theorem (ABC for function fields.) Indeed, you could probably prove in this way that not only are the roots of f\_n(t) not rational, but f\_n(t) has irreducible factors of arbitrarily large degree.


I don't think this approach would touch a harder question along the same lines like [this one](https://mathoverflow.net/questions/5994/number-theoretic-spectral-properties-of-random-graphs/6053#6053). 


