On the Complex Narayana Numbers and its Properties
Kübra Yalvaç *
Tatvan Türkiye Century IMKB Middle School, Bitlis, Turkey.
Hasan Gökbaş
Mathematics Department, Science and Arts Faculty, Bitlis Eren University, Bitlis, Turkey.
*Author to whom correspondence should be addressed.
Abstract
Background: The Narayana sequence is a mathematical sequence based on a delayed reproduction model, extending concepts similar to the Fibonacci sequence and studied through its recurrence relations and related properties.
Aims: This study aims to define the complex Narayana number sequence and to investigate its fundamental properties.
Study Design: This research is theoretical and analytical in nature. Based on a comprehensive review of the literature on Narayana and related number sequences, a complex extension of the Narayana sequence is introduced. Its recurrence relation and initial conditions are determined, and the characteristic equation is obtained. By analyzing the roots, the Binet formula is derived. In addition, the generating function and matrix representation of the sequence are established. Positive and negative indexed terms are examined, and the validity of Catalan, Cassini, and D’Ocagne-type identities is investigated through analytical proof techniques.
Place and Duration of Study: This thesis study was conducted at the Faculty of Science and Letters, Bitlis Eren University. The study did not involve any experimental or applied processes; it was carried out within a theoretical framework based on literature review and analytical methods. The research was completed between November 2025 and February 2026.
Methodology: The study was conducted using theoretical and analytical research methods.
The relevant literature has been thoroughly reviewed; existing studies on the Narayana number sequence and related number sequences have been examined. The complex Narayana number sequence has been defined, and its recurrence relation and initial conditions have been determined. The characteristic equation was obtained, and Binet's formula was derived by analyzing its roots. The sequence generator function and matrix representation were created using analytical methods. Terms with positive and negative indices have been calculated and presented in a table. The adaptability of Catalan, Cassini, and D'Ocagne type identities to the complex Narayana number sequence has been investigated, and the necessary proofs have been provided. All results obtained have been verified using algebraic transformations, mathematical inferences, and analytical proof techniques.
Results: The study successfully defines the complex Narayana number sequence and establishes its main structural properties. Explicit forms such as the Binet formula, generating function, and matrix representation are obtained. Furthermore, it is shown that classical identities, including Catalan, Cassini, and D’Ocagne-type identities, can be extended to the complex Narayana number sequence. These results are rigorously verified using algebraic and analytical methods.
Conclusion: The findings contribute to the generalization of classical number sequences into the complex domain and provide a comprehensive framework for the analysis of the complex Narayana number sequence. This study is expected to support further research on generalized recursive sequences and their algebraic properties.
Keywords: Sequences numbers, complex numbers, Narayana numbers, gauss Narayana numbers