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q-Fibonacci Polynomials and the Rogers-Ramanujan Identities
Johann Cigler
Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria
johann.cigler@univie.ac.at
Annals of Combinatorics 8 (3) p.269-285 September, 2004
AMS Subject Classification:05A19, 05A30, 11B39, 11B65, 11P81
Abstract:
We derive some formulas for the Carlitz q-Fibonacci polynomials Fn(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 R2 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.
Keywords: Rogers-Ramanujan identities, q-analogue, q-Fibonacci polynomial, lattice path, inclusion-exclusion

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