On Dual Hyperbolic Narayana and Narayana-Lucas Hybrid Quaternions

Pankaj Kumar; Shilpa Kapoor

doi:10.32323/ujma.1686437

On Dual Hyperbolic Narayana and Narayana-Lucas Hybrid Quaternions

Abstract

In this present communication, we introduce the novel concepts of dual hyperbolic Narayana quaternions and dual hyperbolic Narayana-Lucas quaternions within the framework of hybrid numbers. We also explore the connections between these newly defined quaternions and examine the mathematical properties they share. Additionally, we find the recurrence relations, Binet formulas, generating functions, exponential generating functions, and other meaningful identities. These newly introduced quaternions have significant applications in the fields of quantum physics, computer graphics, and robotics. Additionally, these identities and relationships we established also play an important role in the field of number theory and combinatorics.

Keywords

Hybrid number, Hyperbolic, Narayana-Lucas sequence, Quaternions

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Details

Primary Language

English

Subjects

Algebra and Number Theory

Journal Section

Research Article

Authors

Pankaj Kumar ^*
0000-0002-2938-1033
India

Shilpa Kapoor
0000-0001-9842-6226
India

Early Pub Date

September 5, 2025

Publication Date

September 17, 2025

Submission Date

April 29, 2025

Acceptance Date

July 13, 2025

Published in Issue

Year 2025 Volume: 8 Number: 3

DOI

https://doi.org/10.32323/ujma.1686437

IZ

https://izlik.org/JA23NY93YX

APA

Kumar, P., & Kapoor, S. (2025). On Dual Hyperbolic Narayana and Narayana-Lucas Hybrid Quaternions. Universal Journal of Mathematics and Applications, 8(3), 116-125. https://doi.org/10.32323/ujma.1686437

AMA

1.Kumar P, Kapoor S. On Dual Hyperbolic Narayana and Narayana-Lucas Hybrid Quaternions. Univ. J. Math. Appl. 2025;8(3):116-125. doi:10.32323/ujma.1686437

Chicago

Kumar, Pankaj, and Shilpa Kapoor. 2025. “On Dual Hyperbolic Narayana and Narayana-Lucas Hybrid Quaternions”. Universal Journal of Mathematics and Applications 8 (3): 116-25. https://doi.org/10.32323/ujma.1686437.

EndNote

Kumar P, Kapoor S (September 1, 2025) On Dual Hyperbolic Narayana and Narayana-Lucas Hybrid Quaternions. Universal Journal of Mathematics and Applications 8 3 116–125.

IEEE

[1]P. Kumar and S. Kapoor, “On Dual Hyperbolic Narayana and Narayana-Lucas Hybrid Quaternions”, Univ. J. Math. Appl., vol. 8, no. 3, pp. 116–125, Sept. 2025, doi: 10.32323/ujma.1686437.

ISNAD

Kumar, Pankaj - Kapoor, Shilpa. “On Dual Hyperbolic Narayana and Narayana-Lucas Hybrid Quaternions”. Universal Journal of Mathematics and Applications 8/3 (September 1, 2025): 116-125. https://doi.org/10.32323/ujma.1686437.

JAMA

1.Kumar P, Kapoor S. On Dual Hyperbolic Narayana and Narayana-Lucas Hybrid Quaternions. Univ. J. Math. Appl. 2025;8:116–125.

MLA

Kumar, Pankaj, and Shilpa Kapoor. “On Dual Hyperbolic Narayana and Narayana-Lucas Hybrid Quaternions”. Universal Journal of Mathematics and Applications, vol. 8, no. 3, Sept. 2025, pp. 116-25, doi:10.32323/ujma.1686437.

Vancouver

1.Pankaj Kumar, Shilpa Kapoor. On Dual Hyperbolic Narayana and Narayana-Lucas Hybrid Quaternions. Univ. J. Math. Appl. 2025 Sep. 1;8(3):116-25. doi:10.32323/ujma.1686437