On Generalized Third-Order Jacobsthal Numbers

Abstract

In this paper, we investigate the generalized third order Jacobsthal sequences and we deal with, in detail, four special cases, namely, third order Jacobsthal, third order Jacobsthal-Lucas, modified third order Jacobsthal, third order Jacobsthal Perrin sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

Matrix Structure of Jacobsthal Numbers

“…Following the authors in [2,5,[20][21][22][23][24], we define, for n > 0 , 1 ≤ r ≤ j, and s ≥ 0, the j sequences of the generalized order j -Jacobsthal numbers as follows:…”

Generalized Friedrich Numbers

“…respectively, see Soykan [4] for more details. So, by using Binet's formulas of Friedrich, Friedrich-Lucas and third-order Jacobsthal, modified third-order Jacobsthal, third-order Jacobsthal-Lucas numbers, (or by using mathematical induction), we get the following Lemma which contains many identities:…”

“…The sequences {Jn} n≥0 and {jn} n≥0 are defined in [2] and {Kn} n≥0 is given in [3]. For more details on the generalized third-order Jacobsthal numbers and its special cases, see [4].…”

Jacobsthal Matrices and their Properties

“…E.Andrade and D.C Olivera, C. Manzaneda introduced the concept on circulant like matrices properties involving Horadam, Fibonacci, Jacobsthal and Pell numbers (4) . E.E Polatli and Y Soykan introduced the concept of third -order Jacobsthal sequence and third-order Jacobsthal-Lucas sequence (5) . D Brod, A S Lina and Iwona Wloch introduced the concept of two generalizations of dual-hyperbolic balancing numbers (6) .…”