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q-Fibonacci Polynomials and the Rogers-Ramanujan Identities
Johann Cigler
Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria
Annals of Combinatorics 8 (3) p.269-285 September, 2004
AMS Subject Classification:05A19, 05A30, 11B39, 11B65, 11P81
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


1. A.K. Agarwal and D.M. Bressoud, Lattice paths and multiple basic hypergeometric series, Pacific J. Math. 136 (1989) 209--228.

2. G.E. Andrews, A polynomial identity which implies the Rogers-Ramanujan identities, Scripta Math. 28 (1970) 297--305.

3. G.E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.

4. G.E. Andrews, On the proofs of the Rogers-Ramanujan identities, In: q-Series and Partitions, The IMA Volumes in Mathematics and Its Applications 18, Springer-Verlag, New York, 1989, pp. 1--14

5. D.M. Bressoud, An easy proof of the Rogers-Ramanujan identities, J. Number Theory 16 (1983) 235--241.

6. D.M. Bressoud, Lattice paths and the Rogers-Ramanujan identities, Lecture Notes in Mathematics 1395, 1987, pp. 140--172.

7. W.H. Burge, Restricted partition pairs, J. Combin. Theory, Ser. A 63 (1993) 210--222.

8. L. Carlitz, Fibonacci notes 4: q-Fibonacci polynomials, Fibonacci Quart. 13 (1975) 97--102.

9. J. Cigler, q-Fibonacci polynomials, Fibonacci Quart. 41 (2003) 31--40.

10. C. Krattenthaler and S.G. Mohanty, On lattice path counting by major and descents, Europ. J. Combin. 14 (1993) 43--51.

11. S.G. Mohanty, Lattice Path Counting and Applications, Academic Press, 1979.

12. M. Okado, A. Schilling, and M. Shimozono, Crystal bases and q-identities, Contemp. Math. 291 (2001) 29--53.

13. P. Paule, The concept of Bailey chains, Sém. Lothar. Combin. B18f (1987) 1--24.

14. P. Paule, Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type, Elect. J. Combin. 1 (1994) #R10.

15. H. Prodinger, Lecture Notes on the Course q-Series in Combinatorics and Number Theory, http://www.wits.ac.za/helmut/postscriptfiles/q course.ps.gz.

16. A. Schilling and S.O. Warnaar, Conjugate Bailey pairs, Contemp. Math. 297 (2002) 227--255.

17. A.V. Sills, Finite Rogers-Ramanujan type identities, Elect. J. Combin. 10 (2003) #R13.

18. I. Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche, 1917, In: Gesammelte Abhandlungen, Bd.2, Springer-Verlag, 1973, pp. 117--136.