On the combinatorics of polynomial generalizations of Rogers–Ramanujan-type identities

Abstract

In this paper following some ideas introduced by Andrews (Combinatorics and Ramanujan's “lost” notebook, London Mathametical Society Lecture Note Series, No. 103, Cambridge University Press, London, 1985, pp. 1–23) and results given by Santos (Computer algebra and identities of the Rogers–Ramanujan type. Ph.D. Thesis, Pennsylvania State University, 1991) we give a polynomial generalization for the Fibonacci sequence from which we get new formula and combinatorial interpretation for the Fibonacci numbers.