  The aim of this paper is the study of $q^{-1}$-Fibonacci polynomials with
$0<q<1$. First, the $q^{-1}$-Fibonacci polynomials are related to a
$q$-exponential function which allows an asymptotic analysis to be worked out.
Second, related basic orthogonal polynomials are investigated with the emphasis
on their orthogonality properties. In particular, a compact formula for the
reproducing kernel is obtained that allows to describe all the N-extremal
measures of orthogonality in terms of basic hypergeometric functions and their
zeros. Two special cases involving $q$-sine and $q$-cosine are discussed in
more detail.
