A Study of Dual-Hyperbolic Third-Order $k$-Jacobsthal Numbers

Gamaliel Morales

doi:10.33401/fujma.1783934

A Study of Dual-Hyperbolic Third-Order $k$-Jacobsthal Numbers

Abstract

In this study, we introduce one-parameter generalization of dual-hyperbolic third-order Jacobsthal (or dual-hyperbolic third-order $k$-Jacobsthal) numbers. We present some identities and properties of them, among others the Binet-type formula, d'Ocagne and Cassini identities. Furthermore, we study the summation formula and generating function for these dual-hyperbolic numbers. The results presented here are a generalizations of the results for the dual-hyperbolic Jacobsthal numbers of order two. New identities for this sequence including its matrix representation are introduced.

Keywords

References

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Details

Primary Language

English

Subjects

Algebra and Number Theory

Journal Section

Research Article

Authors

Gamaliel Morales ^*
0000-0003-3164-4434
Chile

Publication Date

March 30, 2026

Submission Date

September 14, 2025

Acceptance Date

February 11, 2026

Published in Issue

Year 2026 Volume: 9 Number: 1

DOI

https://doi.org/10.33401/fujma.1783934

IZ

https://izlik.org/JA22CL64CC

Cite

RIS / Bibtex

APA

Morales, G. (2026). A Study of Dual-Hyperbolic Third-Order $k$-Jacobsthal Numbers. Fundamental Journal of Mathematics and Applications, 9(1), 23-30. https://doi.org/10.33401/fujma.1783934

AMA

1.Morales G. A Study of Dual-Hyperbolic Third-Order $k$-Jacobsthal Numbers. Fundam. J. Math. Appl. 2026;9(1):23-30. doi:10.33401/fujma.1783934

Chicago

Morales, Gamaliel. 2026. “A Study of Dual-Hyperbolic Third-Order $k$-Jacobsthal Numbers”. Fundamental Journal of Mathematics and Applications 9 (1): 23-30. https://doi.org/10.33401/fujma.1783934.

EndNote

Morales G (March 1, 2026) A Study of Dual-Hyperbolic Third-Order $k$-Jacobsthal Numbers. Fundamental Journal of Mathematics and Applications 9 1 23–30.

IEEE

[1]G. Morales, “A Study of Dual-Hyperbolic Third-Order $k$-Jacobsthal Numbers”, Fundam. J. Math. Appl., vol. 9, no. 1, pp. 23–30, Mar. 2026, doi: 10.33401/fujma.1783934.

ISNAD

Morales, Gamaliel. “A Study of Dual-Hyperbolic Third-Order $k$-Jacobsthal Numbers”. Fundamental Journal of Mathematics and Applications 9/1 (March 1, 2026): 23-30. https://doi.org/10.33401/fujma.1783934.

JAMA

1.Morales G. A Study of Dual-Hyperbolic Third-Order $k$-Jacobsthal Numbers. Fundam. J. Math. Appl. 2026;9:23–30.

MLA

Morales, Gamaliel. “A Study of Dual-Hyperbolic Third-Order $k$-Jacobsthal Numbers”. Fundamental Journal of Mathematics and Applications, vol. 9, no. 1, Mar. 2026, pp. 23-30, doi:10.33401/fujma.1783934.

Vancouver

1.Gamaliel Morales. A Study of Dual-Hyperbolic Third-Order $k$-Jacobsthal Numbers. Fundam. J. Math. Appl. 2026 Mar. 1;9(1):23-30. doi:10.33401/fujma.1783934