  Fibonacci polynomials are generalizations of Fibonacci numbers, so it is
natural to consider polynomial versions of the various results for Fibonacci
numbers. According to Hong, Pongsriiam, Bulawa, and Lee, the generating
function of the Fibonacci sequence in the domain of rational numbers,
$f(t)=t/(1-t-t^2)$, takes an integer value if and only if $t=F_{k}/F_{k+1}$ for
some $k \in \N$ or $t=-F_{k+1}/F_{k}$ for some $k \in \N^{+}$, where $F_{k}$ is
the $k$th Fibonacci number. This study is built upon their work by considering
polynomial sequences that satisfy the recurrence relation
$F_{i+2}(x)=axF_{i+1}(x)+bF_{i}(x)$ with initial values $(F_{0}(x),
F_{1}(x))=(0, 1)$, where $a$ and $b$ are positive integers such that $b|a$. As
an application, for a square-free natural number $d \in \N$, we verify the
results are of the same form as the above for the generating function of the
sequence satisfying the recurrence relation $F_{i+2}(\sqrt{d})=a\sqrt{d}
F_{i+1}(\sqrt{d})+bF_{i}(\sqrt{d})$ with initial values $(F_{0}(\sqrt{d}),
F_{1}(\sqrt{d}))=(0, 1)$.
