q-Fibonacci Polynomials and the Rogers-Ramanujan Identities

Abstract.

We derive some formulas for the Carlitz q-Fibonacci polynomials F _n (t) which reduce to the finite version of the Rogers-Ramanujan identities obtained by I. Schur for t = 1. Our starting point is a representation of the q-Fibonacci polynomials as the weight of certain lattice paths in $\mathbb{R}^2$ contained in a strip along the x-axis. We give an elementary combinatorial proof by using only the principle of inclusion-exclusion and some standard facts from q-analysis.