A Detailed Study on Matrix Sequence of Generalized Narayana Numbers

Abstract

In this paper, we introduce and investigate the generalized Narayana matrix sequence and we deal with, in detail, three special cases of this sequence which we call them Narayana, Narayana-Lucas and Narayana-Perrin matrix sequences. We present Binet’s formulas, generating functions, and the summation formulas for these sequences. We present the proofs to indicate how these sum formulas, in general, were discovered. Of course, all the listed sum formulas may be proved by induction, but that method of proof gives no clue about their discovery. Moreover, we give some identities and matrices related with these sequences. Furthermore, we show that there always exist interrelation between generalized Narayana, Narayana, Narayana-Lucas and Narayana-Perrin matrix sequences.