Binomial Transforms of k-Narayana Sequences and Some Properties

Authors : Faruk Kaplan, Arzu Özkoç Öztürk

Pages : 137-146

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Publication Date : 2024-09-30

Article Type : Research

Abstract :The aim of the study is to obtain new binomial transforms for the $k-$ Narayana sequence. The first of these is the binomial transform, which is its normal form, and in the first step, after finding the recurrence relation of this new binomial transform, the generating function and Binet formula were obtained. Finally, Pascal\'s triangle was calculated. In the rest of the article, $k-$binomial transform was performed for the $k-$ Narayana sequence and the recurrence relation, generating function, Binet formula and Pascal\'s triangle were examined for the new sequence obtained. Then, by performing the falling binomial transform and the rising binomial transform, the features listed above were found again for these sequences.
Keywords : $k-$Narayana Sequences, Binomial transform, Recurrence relations, Binet formulas