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An Application of Sobolev Orthogonal Polynomials to the Computation of a Special Hankel Determinant

  • Paul Barry
  • Predrag M. Rajković
  • Marko D. Petković
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 42)

Abstract

Many Hankel determinant computations arising in combinatorial analysis can be done using results from the theory of standard orthogonal polynomials. Here, we will emphasize special sequences which require the inclusion of discrete Sobolov orthogonality to find their closed form.

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Notes

Acknowledgements

This research was supported by the Science Foundation of Republic Serbia, Project No. 144023 and Project No. 144011.

References

  1. 1.
    Barry, P.: A Catalan transform and related transformations on integer sequences. J. Integer Seq. 8, Article 05.4.5 (2005)Google Scholar
  2. 2.
    Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)MATHGoogle Scholar
  3. 3.
    Cvetković, A., Rajković, P., Ivković, M.: Catalan numbers, the Hankel transform and Fibonacci numbers. J. Integer Seq. 5, Article 02.1.3 (2002)Google Scholar
  4. 4.
    Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Clarendon, Oxford (2004)MATHGoogle Scholar
  5. 5.
    Krattenthaler, C.: Advanced determinant calculus: A complement. Linear Algebra Appl. 411 (2005) 68–166MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Layman, J.W.: The Hankel transform and some of its properties. J. Integer Seq. 4, Article 01.1.5 (2001)Google Scholar
  7. 7.
    Marcellán, F., Ronveaux, A.: On a class of polynomials orthogonal with respect to a discrete Sobolev inner product. Indag. Mathem. N.S. 1, 451–464 (1990)Google Scholar
  8. 8.
    Rajković, P.M., Petković, M.D., Barry, P.: The Hankel transform of the sum of consecutive generalized Catalan numbers. Integral Transform. Spec. Funct. 18 No. 4, 285–296 (2007)Google Scholar
  9. 9.
    Sloane, NJA: The On-Line Encyclopedia of Integer Sequences. Published electronically at http://www.research.att.com/~njas/sequences/, 2007

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Paul Barry
    • 1
  • Predrag M. Rajković
    • 2
  • Marko D. Petković
    • 3
  1. 1.School of ScienceWaterford Institute of TechnologyWaterfordIreland
  2. 2.Faculty of Mechanical EngineeringUniversity of NišNišSerbia
  3. 3.Faculty of Sciences and MathematicsUniversity of NišNišSerbia

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