Skip to main content
SpringerLink
Log in

Closed-form expression for Hankel determinants of the Narayana polynomials

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

We considered a Hankel transform evaluation of Narayana and shifted Narayana polynomials. Those polynomials arises from Narayana numbers and have many combinatorial properties. A mainly used tool for the evaluation is the method based on orthogonal polynomials. Furthermore, we provided a Hankel transform evaluation of the linear combination of two consecutive shifted Narayana polynomials, using the same method (based on orthogonal polynomials) and previously obtained moment representation of Narayana and shifted Narayana 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. P. Barry: On integer-sequences-based constructions of generalized Pascal triangles. J. Integer Seq. 59 (2006). Article 06.2.4. Electronic only.

  2. P. Barry, A. Hennessy: Notes on a family of Riordan arrays and associated integer Hankel transforms. J. Integer Seq. 12 (2009). Article ID 09.5.3. Electronic only.

  3. P. Brändén: q-Narayana numbers and the flag h-vector of J(2×n). Discrete Math. 281 (2004), 67–81.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. A. Brualdi, S. Kirkland: Aztec diamonds and digraphs, and Hankel determinants of Schröder numbers. J. Comb. Theory Ser. B 94 (2005), 334–351.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Chamberland, C. French: Generalized Catalan numbers and generalized Hankel transformations. J. Integer Seq. 10 (2007).

  6. T. S. Chihara: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York, 1978.

    MATH  Google Scholar 

  7. A. Cvetković, P. M. Rajković, M. Ivković: Catalan Numbers, the Hankel transform, and Fibonacci numbers. J. Integer Seq. 5 (2002).

  8. O. Eğecioğlu, T. Redmond, C. Ryavec: Almost product evaluation of Hankel determinants. Electron. J. Comb. 15,#R6 (2008).

  9. O. Eğecioğlu, T. Redmond, C. Ryavec: A multilinear operator for almost product evaluation of Hankel determinants. J. Comb. Theory Ser. A 117 (2010), 77–103.

    Article  MATH  Google Scholar 

  10. M. Garcia Armas, B.A. Sethuraman: A note on the Hankel transform of the central binomial coefficients. J. Integer Seq. 11 (2008). Article ID 08.5.8.

  11. W. Gautschi: Orthogonal polynomials: Applications and computation. In: Acta Numerica Vol. 5 (A. Iserles, ed.). Cambridge University Press, Cambridge, 1996, pp. 45–119.

    Google Scholar 

  12. W. Gautschi: Orthogonal Polynomials. Computation and Approximation. Oxford University Press, Oxford, 2004.

    MATH  Google Scholar 

  13. M. E. H. Ismail: Determinants with orthogonal polynomial entries. J. Comput. Appl. Math. 178 (2005), 255–266.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Junod: Hankel determinants and orthogonal polynomials. Expo. Math. 21 (2003), 63–74.

    Article  MathSciNet  MATH  Google Scholar 

  15. C. Krattenthaler: Advanced determinant calculus: A complement. Linear Algebra Appl. 411 (2005), 68–166.

    Article  MathSciNet  MATH  Google Scholar 

  16. W. Lang: On sums of powers of zeros of polynomials. J. Comput. Appl. Math. 89 (1998), 237–256.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. W. Layman: The Hankel transform and some of its properties. J. Integer Seq. 4 (2001). Article 01.1.5. Electronic only.

  18. L. L. Liu, Y. Wang: A unified approach to polynomial sequences with only real zeros. Adv. Appl. Math. 38 (2007), 542–560.

    Article  MATH  Google Scholar 

  19. P. A. MacMahon: Combinatorial Analysis, Vols. 1 and 2. Cambridge University Press, Cambridge, 1915, 1916, reprinted by Chelsea, 1960.

    Google Scholar 

  20. T. Mansour, Y. Sun: Identities involving Narayana polynomials and Catalan numbers. Discrete Math. 309 (2009), 4079–4088.

    Article  MathSciNet  MATH  Google Scholar 

  21. T. V. Narayana: Sur les treillis formés par les partitions d’un entier et leurs applications à la théorie des probabilités. C. R. Acad. Sci. 240 (1955), 1188–1189. (In French.)

    MATH  Google Scholar 

  22. P. M. Rajković, M.D. Petković, P. Barry: The Hankel transform of the sum of consecutive generalized Catalan numbers. Integral Transforms Spec. Funct. 18 (2007), 285–296.

    Article  MathSciNet  MATH  Google Scholar 

  23. N. J. A. Sloane: The On-Line Encyclopedia of Integer Sequences; available at http://www.research.att.com/~njas/sequences.

  24. R. A. Sulanke: Counting lattice paths by Narayana polynomials. Electron. J. Comb. 7 (2000). #R40.

    Google Scholar 

  25. R. A. Sulanke: The Narayana distribution. J. Statist. Plann. Inference 101 (2002), 311–326.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Xin: Proof of the Somos-4 Hankel determinants conjecture. Adv. Appl. Math. 42 (2009), 152–156.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000, Niš, Serbia

    Marko D. Petković

  2. School of Science, Waterford Institute of Technology, Waterford, Ireland

    Paul Barry

  3. Faculty of Mechanical Engineering, University of Niš, Aleksandra Medvedeva 14, 18000, Niš, Serbia

    Predrag Rajković

Authors
  1. Marko D. Petković
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul Barry
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Predrag Rajković
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marko D. Petković.

Additional information

This research was supported by the Science Foundation of Republic Serbia, Project No. 174011 and Project No. 174013.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Petković, M.D., Barry, P. & Rajković, P. Closed-form expression for Hankel determinants of the Narayana polynomials. Czech Math J 62, 39–57 (2012). https://doi.org/10.1007/s10587-012-0015-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10587-012-0015-8

Keywords

MSC 2010

Search

Navigation