On the Properties of Iterated Binomial Transforms for the Padovan and Perrin Matrix Sequences

Abstract

In this study, we apply "r" times the binomial transform to the Padovan and Perrin matrix sequences. Also, the Binet formulas, summations, generating functions of these transforms are found using recurrence relations. Finally, we give the relationships of between iterated binomial transforms for Padovan and Perrin matrix sequences.

Padovan and Perrin generalized quaternions

“…Proof. By using Definition 1 and Equations ( 5), ( 29), (31), and (18a)-(18i), the proofs are clear. Additionally, by using the Theorem 9, the alternative proofs can be provided (e.g., by replacing n → 2n, m → 2m in part ii.…”

“…Proof. The proofs are straightforward using Definition 1 and Equations ( 5), ( 29), (31), and (14h)-(14k).…”

“…Proof. The proofs are obvious from Definition 1 and Equations ( 5), ( 29), (31), and (17a)-(17d). Alternatively, Theorem 9 can be used (e.g., by substituting n → 2n, m → 2m in part i.…”

“…[11][12][13][14][15][16][17][18][19][20] The master study is undertaken by Shannon et al 19 Matrix representations and matrix sequences of Padovan and Perrin numbers are studied by several authors. [21][22][23][24][25][26][27][28][29][30][31] Furthermore, a great number of studies have been carried out with regard to quaternions with special number components. Halıcı, 32 Horadam, 33 and İyer 34 are examined Fibonacci and Lucas quaternions.…”

On the binomial transforms of the Horadam quaternion sequences

“…Falcon, 13 Yılmaz and Taşkara 14 apply “ r ” times the binomial transform to the k ‐Fibonacci Sequence and the Padovan and Perrin Matrix Sequences, respectively. Also, they represent some specialty of these iterated binomial transforms.…”