Padovan and Perrin Hyperbolic Spinors

TL;DR

This paper introduces Padovan and Perrin hyperbolic spinors derived from third-order recurrence sequences and split quaternions, establishing new mathematical relationships and formulas with potential applications across disciplines.

Contribution

It develops the concept of Padovan and Perrin hyperbolic spinors, providing formulas, algorithms, and generalizations that connect number theory, hyperbolic spinors, and split quaternions.

Findings

Defined new Padovan and Perrin hyperbolic spinors

Derived Binet, generating, and summation formulas

Presented matrix and determinant equations

Abstract

In this study, we intend to bring together Padovan and Perrin number sequences, which are one of the most popular third-order recurrence sequences, and hyperbolic spinors, which are used in several disciplines from physics to mathematics, with the help of the split quaternions. This paper especially improves the relationship between hyperbolic spinors both a physical and mathematical concept, and number theory. For this aim, we combine the hyperbolic spinors and Padovan and Perrin numbers concerning the split Padovan and Perrin quaternions, and we determine two new special recurrence sequences named Padovan and Perrin hyperbolic spinors. Then, we give Binet formulas, generating functions, exponential generating functions, Poisson generating functions, and summation formulas. Additionally, we present some matrix and determinant equations with respect to them. Then, we construct some numerical algorithms for these special number systems, as well. Further, we give an introduction for $(s,t)$-Padovan and $(s,t)$-Perrin hyperbolic spinors.