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

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...

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Bibliographic Details
Published inBulletin of the Malaysian Mathematical Sciences Society Vol. 40; no. 1; pp. 19 - 33
Main Authors Bojičić, Radica, Petković, Marko D.
Format Journal Article
LanguageEnglish
Published Singapore Springer Singapore 2017
Springer Nature B.V
Subjects
Applications of Mathematics
Closed form solutions
Exact solutions
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
Hankel transform
Secondary 11C20
11Y55
Primary 11B83
Motzkin numbers
Orthogonal polynomials
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Summary: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.
ISSN:0126-6705
2180-4206
DOI:10.1007/s40840-015-0249-3