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By taking a limit of (23) when c → −1, we find that the Hankel transform of m n+2 + 2m n+1 + m n is given byĥ n (−1) = (30 + 37n + 15n 2 + 2n 3 )/6. The same sequence is obtained in the case c → 3
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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.
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Oct 31, 2015
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