Skip to main content
SpringerLink
Log in

New hypergeometric connection formulae between Fibonacci and Chebyshev polynomials

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We establish new connection formulae between Fibonacci polynomials and Chebyshev polynomials of the first and second kinds. These formulae are expressed in terms of certain values of hypergeometric functions of the type \(_2F_{1}\). Consequently, we obtain some new expressions for the celebrated Fibonacci numbers and their derivative sequences. Moreover, we evaluate some definite integrals involving products of Fibonacci and Chebyshev polynomials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abd-Elhameed, W.M.: New product and linearization formulae of Jacobi polynomials of certain parameters. Integral Transform Spec. Funct. 26(8), 586–599 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abd-Elhameed, W.M., Doha, E.H., Ahmed, H.M.: Linearization formulae for certain Jacobi polynomials. Ramanujan J. doi:10.1007/s11139-014-9668-2

  3. Area, I., Godoy, E., Ronveaux, A., Zarzo, A.: Solving connection and linearization problems within the askey scheme and its \(q\)-analogue via inversion formulas. J. Comput. Appl. Math. 133(1), 151–162 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bozkurt, S.B., Yılmaz, F., Bozkurt, D.: On the complex factorization of the Lucas sequence. Appl. Math. Lett. 24(8), 1317–1321 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dilcher, K., Pisano, L.: Hypergeometric functions and Fibonacci numbers. Fibonacci Quart. 38(4), 342–362 (2000)

    MathSciNet  MATH  Google Scholar 

  6. Doha, E.H.: On the connection coefficients and recurrence relations arising from expansions in series of Laguerre polynomials. J. Phys. A: Math. Gen. 36(20), 5449–5462 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Doha, E.H.: On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials. J. Phys. A: Math. Gen. 37(3), 657 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Doha, E.H., Abd-Elhameed, W.M.: Integrals of Chebyshev polynomials of third and fourth kinds: an application to solution of boundary value problems with polynomial coefficients. J. Contemp. Math. Anal. 49(6), 296–308 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Doha, E.H., Abd-Elhameed, W.M.: New linearization formulae for the products of Chebyshev polynomials of third and fourth kind. Rocky Mt. J. Math. (to appear)

  10. Doha, E.H., Ahmed, H.M.: Recurrences and explicit formulae for the expansion and connection coefficients in series of Bessel polynomials. J. Phys. A: Math. Gen. 37(33), 8045 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. El-Mikkawy, M., Sogabe, T.: A new family of \(k\)-Fibonacci numbers. Appl. Math. Comput. 215(12), 4456–4461 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Falcon, S., Plaza, A.: On the Fibonacci \(k\)-numbers. Chaos Soliton Fract. 32(5), 1615–1624 (2007)

    Article  MATH  Google Scholar 

  13. Gulec, H.H., Taskara, N., Uslu, K.: A new approach to generalized Fibonacci and Lucas numbers with binomial coefficients. Appl. Math. Comput. 220, 482–486 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Koshy, T.: Fibonacci and Lucas numbers with applications, vol. 51. Wiley, New York (2011)

    MATH  Google Scholar 

  15. Maroni, P., da Rocha, Z.: Connection coefficients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation. Numer. Algor. 47(3), 291–314 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman and Hall, New York (2010)

    MATH  Google Scholar 

  17. Sánchez-Ruiz, J., Dehesa, J.S.: Some connection and linearization problems for polynomials in and beyond the askey scheme. J. Comput. Appl. Math. 133(1), 579–591 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Szwarc, R.: Linearization and connection coefficients of orthogonal polynomials. Mh. Math. 113(4), 319–329 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  19. Taskara, N., Uslu, K., Gulec, H.H.: On the properties of Lucas numbers with binomial coefficients. Appl. Math. Lett. 23(1), 68–72 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Włoch, A.: Some identities for the generalized Fibonacci numbers and the generalized Lucas numbers. Appl. Math. Comput. 219(10), 5564–5568 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Yazlik, Y., Taskara, N.: A note on generalized-Horadam sequence. Comput. Math. Appl. 63(1), 36–41 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, W.: On Chebyshev polynomials and Fibonacci numbers. Fibonacci Quart. 40(5), 424–428 (2002)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous referee for his valuable comments which improved the manuscript to its present form.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia

    W. M. Abd-Elhameed

  2. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

    W. M. Abd-Elhameed & Y. H. Youssri

  3. Mathematics and Actuarial Science Department, American University in Cairo, AUC Avenue, New Cairo, Egypt

    Nermine El-Sissi & Mohammad Sadek

Authors
  1. W. M. Abd-Elhameed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. H. Youssri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nermine El-Sissi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mohammad Sadek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to W. M. Abd-Elhameed.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abd-Elhameed, W.M., Youssri, Y.H., El-Sissi, N. et al. New hypergeometric connection formulae between Fibonacci and Chebyshev polynomials. Ramanujan J 42, 347–361 (2017). https://doi.org/10.1007/s11139-015-9712-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-015-9712-x

Keywords

Mathematics Subject Classification

Search

Navigation