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Orthogonal Polynomials Approach to the Hankel Transform of Sequences Based on Motzkin Numbers

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Abstract

In this paper, we use a method based on orthogonal polynomials to give closed-form evaluations of the Hankel transform of sequences based on the Motzkin numbers. It includes linear combinations of consecutive two, three, and four Motzkin numbers. In some cases, we were able to derive the closed-form evaluation of the Hankel transform, while in the others we showed that the Hankel transform satisfies a particular difference equation. As a corollary, we reobtain known results and show some new results regarding the Hankel transform of Motzkin and shifted Motzkin numbers. Those evaluations also give an idea on how to apply the method based on orthogonal polynomials on the sequences having zero entries in their Hankel transform.

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Acknowledgments

Marko D. Petković gratefully acknowledges the support of the research project 174013 of the Serbian Ministry of Education and Science. Authors wish to thank Professor Predrag M. Rajković for useful discussions on this topic.

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Authors and Affiliations

  1. Faculty of Economy, University of Priština, Kosovska Mitrovica, Serbia

    Radica Bojičić

  2. Faculty of Sciences and Mathematics, University of Niš, Niš, Serbia

    Marko D. Petković

Authors
  1. Radica Bojičić
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  2. Marko D. Petković
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Correspondence to Marko D. Petković.

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Communicated by Lee See Keong.

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Bojičić, R., Petković, M.D. Orthogonal Polynomials Approach to the Hankel Transform of Sequences Based on Motzkin Numbers. Bull. Malays. Math. Sci. Soc. 40, 19–33 (2017). https://doi.org/10.1007/s40840-015-0249-3

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  • DOI: https://doi.org/10.1007/s40840-015-0249-3

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