ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 674167, 16 pages
Sequences of Numbers Meet the Generalized Gegenbauer-Humbert Polynomials
1Department of Mathematics and Computer Science, Illinois Wesleyan University, Bloomington, IL 61702, USA
2Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, USA
3Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan
Received 6 July 2011; Accepted 25 August 2011
Academic Editor: W. Liu
Copyright © 2011 Tian-Xiao He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- T. Mansour, “A formula for the generating functions of powers of Horadam's sequence,” Australasian Journal Of Combinatorics, vol. 30, pp. 207–212, 2004.
- A. F. Horadam, “Basic properties of a certain generalized sequence of numbers,” Fibonacci Quarterly, vol. 3, pp. 161–176, 1965.
- L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, The Netherlands, 1974.
- L. C. Hsu, Computational Combinatorics, Shanghai Scientific & Techincal Publishers, Shanghai, China, 1st edition, 1983.
- G. Strang, Linear Algebra and Its Applications, Academic Press/Harcourt Brace Jovanovich Publishers, New York, NY, USA, 2nd edition, 1980.
- H. S. Wilf, Generatingfunctionology, Academic Press, New York, NY, USA, 1990.
- A. T. Benjamin and J. J. Quinn, Proofs that Really Count. The Art of Combinatorial Proof. The Dolciani Mathematical Expositions, vol. 27, Mathematical Association of America, Washington, DC, USA, 2003.
- D. Aharonov, A. Beardon, and K. Driver, “Fibonacci, chebyshev, and orthogonal polynomials,” American Mathematical Monthly, vol. 112, no. 7, pp. 612–630, 2005.
- A. Beardon, “Fibonacci meets Chebyshev,” The Mathematical Gazetle, vol. 91, pp. 251–255, 2007.
- R. B. Marr and G. H. Vineyard, “Five-diagonal Toeplitz determinants and their relation to Chebyshev polynomials,” SIAM Journal on Matrix Analysis and Applications, vol. 9, pp. 579–586, 1988.
- T. X. He and P. J.-S. Shiue, “On sequences of numbers and polynomials defined by linear recurrence relations of order 2,” International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 709386, 21 pages, 2009.
- H. W. Gould, “Inverse series relations and other expansions involving Humbert polynomials,” Duke Mathematical Journal, vol. 32, pp. 697–711, 1965.
- R. Lidl, G. L. Mullen, and G. Turnwald, Dickson Polynomials. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 65, Longman Scientific & Technical/John Wiley & Sons, 1993.
- T. X. He, L. C. Hsu, and P. J.-S. Shiue, “A symbolic operator approach to several summation formulas for power series II,” Discrete Mathematics, vol. 308, no. 16, pp. 3427–3440, 2008.
- L. C. Hsu, On Stirling-Type Pairs and Extended Gegenbauer-Humbert-Fibonacci Polynomials. Applications of Fibonacci Numbers, Vol. 5 (St. Andrews, 1992), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.
- L. C. Hsu and P. J.-S. Shiue, “Cycle indicators and special functions,” Annals of Combinatorics, vol. 5, no. 2, pp. 179–196, 2001.
- W. Y. C. Chen and J. D. Louck, “The combinatorial power of the companion matrix,” Linear Algebra and Its Applications, vol. 232, no. 1–3, pp. 261–278, 1996.
- J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2003.
- G. E. Bergum, L. Bennett, A. F. Horadam, and S. D. Moore, “Jacobsthal polynomials and a conjecture concerning fibonacci-like matrices,” Fibonacci Quarterly, vol. 23, pp. 240–248, 1985.
- H. Civciv and R. Türkmen, “Notes on the (s,t)-Lucas and Lucas matrix sequences,” Ars Combinatoria, vol. 89, pp. 271–285, 2008.