<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>Jacobsthal numbers and Jacobsthal–Lucas numbers are some of the most studied special integer sequences related to the Fibonacci numbers. In this study, we introduce one parameter generalizations of Jacobsthal numbers and Jacobsthal–Lucas numbers. We define two sequences, called generalized Jacobsthal sequence and generalized Jacobsthal–Lucas sequence. We give generating functions, Binet’s formulas for these numbers. Moreover, we obtain some identities, among others Catalan’s, Cassini’s identities and summation formulas for the generalized Jacobsthal numbers and the generalized Jacobsthal–Lucas numbers. These properties generalize the well-known results for classical Jacobsthal numbers and Jacobsthal–Lucas numbers. Additionally, we give a matrix representation of the presented numbers.</p>
</abstract>

Jacobsthal numbers and Jacobsthal–Lucas numbers are some of the most studied special integer sequences related to the Fibonacci numbers. In this study, we introduce one parameter generalizations of Jacobsthal numbers and Jacobsthal–Lucas numbers. We define two sequences, called generalized Jacobsthal sequence and generalized Jacobsthal–Lucas sequence. We give generating functions, Binet’s formulas for these numbers. Moreover, we obtain some identities, among others Catalan’s, Cassini’s identities and summation formulas for the generalized Jacobsthal numbers and the generalized Jacobsthal–Lucas numbers. These properties generalize the well-known results for classical Jacobsthal numbers and Jacobsthal–Lucas numbers. Additionally, we give a matrix representation of the presented numbers.

On Generalized Jacobsthal and Jacobsthal–Lucas Numbers

Faculty of Mathematics and Applied Physics, Department of Discrete Mathematics

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"On Generalized Jacobsthal and Jacobsthal\u2013Lucas Numbers"}

Jacobsthal numbers and Jacobsthal–Lucas numbers are some of the most studied special integer sequences related to the Fibonacci numbers. In this study, we...

On Generalized Jacobsthal and Jacobsthal–Lucas Numbers

Published Online: Jul 04, 2022

Page range: 115 - 128

Received: Dec 05, 2021

Accepted: Jun 16, 2022

DOI: https://doi.org/10.2478/amsil-2022-0011

Keywords
Jacobsthal numbers, Jacobsthal–Lucas numbers, generalized Jacobsthal numbers, Binet formula

© 2022 Dorota Bród et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On Generalized Jacobsthal and Jacobsthal–Lucas Numbers

Published Online: Jul 04, 2022

Page range: 115 - 128

Received: Dec 05, 2021

Accepted: Jun 16, 2022

DOI: https://doi.org/10.2478/amsil-2022-0011

KeywordsJacobsthal numbers, Jacobsthal–Lucas numbers, generalized Jacobsthal numbers, Binet formula

© 2022 Dorota Bród et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Jacobsthal numbers, Jacobsthal–Lucas numbers, generalized Jacobsthal numbers, Binet formula