  The focus of this paper is the study of generalized Fibonacci polynomials and
Fibonomial coefficients. The former are polynomials {n} in variables s and t
given by {0} = 0, {1} = 1, and {n} = s{n-1}+t{n-2} for n ge 2. The latter are
defined by {n choose k} = {n}!/({k}!{n-k}!) where {n}! = {1}{2}...{n}. These
quotients are also polynomials in s and t, and specializations give the
ordinary binomial coefficients, the Fibonomial coefficients, and the q-binomial
coefficients. We present some of their fundamental properties, including a more
general recursion for {n}, an analogue of the binomial theorem, a new proof of
the Euler-Cassini identity in this setting with applications to estimation of
tails of series, and valuations when s and t take on integral values. We also
study a corresponding analogue of the Catalan numbers. Conjectures and open
problems are scattered throughout the paper.
