  In this paper, we develop a new method to produce explicit formulas for the
number $f_{G}(n)$ of rooted spanning forests in the circulant graphs $
G=C_{n}(s_1,s_2,\ldots,s_k)$ and $ G=C_{2n}(s_1,s_2,\ldots,s_k,n).$ These
formulas are expressed through Chebyshev polynomials. We prove that in both
cases the number of rooted spanning forests can be represented in the form
$f_{G}(n)=p\,a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a
prescribed natural number depending on the parity of $n$. Finally, we find an
asymptotic formula for $f_{G}(n)$ through the Mahler measure of the associated
Laurent polynomial $P(z)=2k+1-\sum\limits_{i=1}^k(z^{s_i}+z^{-s_i}).$
