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 $t$. (A weaker version: $F$ should be rational in $x$ for any fixed $t$).

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 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 $a\deg f_n$.)