Generalized Parastichy Fibonacci Numbers and Their Applications

In this study, we define the Parastichy Fibonacci and Parastichy Fibonacci-Lucas numbers. Also, we find basic properties of these numbers, such as characteristic equations and properties. Then we give the Binet formulas, special generating function, and special summation formulas. We associate the terms of the Parastichy Fibonacci and Parastichy Fibonacci-Lucas numbers with matrices. In addition, we calculate the Simson formulas of these numbers. Moreover, we examine the relationships among these numbers. We relate Parastichy Fibonacci numbers with special number sequences such as Lucas numbers. Furthermore, we obtain important correlations between Parastichy Fibonacci and Parastichy Fibonacci-Lucas numbers. We explore the use of generalized Parastichy Fibonacci numbers in the context of hyperbolic quaternions. For these hyperbolic quaternions, we derive various properties, including generating functions and summation formulas. Finally, we establish a relationship between the terms of the Parastichy Fibonacci and Parastichy Fibonacci-Lucas numbers and their associated hyperbolic quaternion values. © The Author(s), under exclusive licence to The National Academy of Sciences, India 2026.