Polynomial Generalizations and Combinatorial Interpretations for Sequences Including the Fibonacci and Pell Numbers

Author(s)
Cecília Pereira de Andrade,
José Plínio de Oliveira Santos,
Elen Viviani Pereira da Silva,
Kênia Cristina Pereira Silva

ABSTRACT

In this paper we present combinatorial interpretations and polynomials generalizations for sequences including the Fibonacci numbers, the Pell numbers and the Jacobsthal numbers in terms of partitions. It is important to mention that results of this nature were given by Santos and Ivkovic in two papers published on the Fibonacci Quarterly, Polynomial generalizations of the Pell sequence and the Fibonacci sequence [1] and Fibonacci Numbers and Partitions [2] , and one, by Santos, on Discrete Mathematics, On the Combinatorics of Polynomial generalizations of Rogers-Ramanujan Type Identities [3]. By these results one can see that from the q-series identities important combinatorial information can be obtained by a careful study of the two variable function introduced by Andrews in Combinatorics and Ramanujan's lost notebook [4].

In this paper we present combinatorial interpretations and polynomials generalizations for sequences including the Fibonacci numbers, the Pell numbers and the Jacobsthal numbers in terms of partitions. It is important to mention that results of this nature were given by Santos and Ivkovic in two papers published on the Fibonacci Quarterly, Polynomial generalizations of the Pell sequence and the Fibonacci sequence [1] and Fibonacci Numbers and Partitions [2] , and one, by Santos, on Discrete Mathematics, On the Combinatorics of Polynomial generalizations of Rogers-Ramanujan Type Identities [3]. By these results one can see that from the q-series identities important combinatorial information can be obtained by a careful study of the two variable function introduced by Andrews in Combinatorics and Ramanujan's lost notebook [4].

Cite this paper

C. Andrade, J. Santos, E. Silva and K. Silva, "Polynomial Generalizations and Combinatorial Interpretations for Sequences Including the Fibonacci and Pell Numbers,"*Open Journal of Discrete Mathematics*, Vol. 3 No. 1, 2013, pp. 25-32. doi: 10.4236/ojdm.2013.31006.

C. Andrade, J. Santos, E. Silva and K. Silva, "Polynomial Generalizations and Combinatorial Interpretations for Sequences Including the Fibonacci and Pell Numbers,"

References

[1] J. P. O. Santos and M. Ivkovic, “Polynomial Generalizations of the Pell Sequence and the Fibonacci Sequence,” The Fibonacci Quarterly, Vol. 43, No. 4, 2005, pp. 328-338.

[2] J. P. O. Santos and M. Ivkovic, “Fibonacci Numbers and Partitions,” The Fibonacci Quarterly, Vol. 41, No. 3, 2003, pp. 263-278.

[3] J. P. O. Santos, “On the Combinatorics of Polynomial Generalizations of Rogers-Ramanujan-Type Identities,” Discrete Mathematics, Vol. 254, No. 1-3, 2002, pp. 497-511. doi:10.1016/S0012-365X(01)00378-8

[4] G. E. Andrews, “Combinatorics and Ramanujan’s ‘Lost’ Notebook,” In: London Mathematical Society Lecture Note Series, No. 103, Cambridge University Press, London, 1985, pp. 1-23.

[5] J. P. O. Santos, “Computer Algebra and Identities of the Rogers-Ramanujan Type,” Ph.D. Thesis, Pennsylvania State University, University Park, 1991.

[6] L. J. Slater, “Further Identities of the Rogers-Ramanujan Type,” Proceedings London Mathematical Society, Vol. s2-54, No. 1, 1952, pp. 147-167. doi:10.1112/plms/s2-54.2.147

[7] L. J. Slater, “A New Proof of Roger’s Transformations of Infinite Series,” Proceedings London Mathematical Society, Vol. s2-53, No. 1, 1951, pp. 460-475. doi:10.1112/plms/s2-53.6.460

[8] A. V. Sills. “RRtools—A Maple Package for Aiding the Discovery and Proof of Finite Rogers-Ramanujan Type Identities,” Journal of Symbolic Computation, Vol. 37, No. 4, 2004, pp. 415-448. http://math.georgiasouthern.edu/asills/maple/RRtools1

[1] J. P. O. Santos and M. Ivkovic, “Polynomial Generalizations of the Pell Sequence and the Fibonacci Sequence,” The Fibonacci Quarterly, Vol. 43, No. 4, 2005, pp. 328-338.

[2] J. P. O. Santos and M. Ivkovic, “Fibonacci Numbers and Partitions,” The Fibonacci Quarterly, Vol. 41, No. 3, 2003, pp. 263-278.

[3] J. P. O. Santos, “On the Combinatorics of Polynomial Generalizations of Rogers-Ramanujan-Type Identities,” Discrete Mathematics, Vol. 254, No. 1-3, 2002, pp. 497-511. doi:10.1016/S0012-365X(01)00378-8

[4] G. E. Andrews, “Combinatorics and Ramanujan’s ‘Lost’ Notebook,” In: London Mathematical Society Lecture Note Series, No. 103, Cambridge University Press, London, 1985, pp. 1-23.

[5] J. P. O. Santos, “Computer Algebra and Identities of the Rogers-Ramanujan Type,” Ph.D. Thesis, Pennsylvania State University, University Park, 1991.

[6] L. J. Slater, “Further Identities of the Rogers-Ramanujan Type,” Proceedings London Mathematical Society, Vol. s2-54, No. 1, 1952, pp. 147-167. doi:10.1112/plms/s2-54.2.147

[7] L. J. Slater, “A New Proof of Roger’s Transformations of Infinite Series,” Proceedings London Mathematical Society, Vol. s2-53, No. 1, 1951, pp. 460-475. doi:10.1112/plms/s2-53.6.460

[8] A. V. Sills. “RRtools—A Maple Package for Aiding the Discovery and Proof of Finite Rogers-Ramanujan Type Identities,” Journal of Symbolic Computation, Vol. 37, No. 4, 2004, pp. 415-448. http://math.georgiasouthern.edu/asills/maple/RRtools1