k-Fibonacci and k-Lucas Differential Equations with New Spirals

Document Type : Original Scientific Paper

Authors

‎Mechanical Engineering Department, Manisa Celal Bayar University, ‎Muradiye‎, ‎Yunusemre‎, ‎Manisa‎, ‎Turkey

10.22052/mir.2026.257640.1541

Abstract

‎Fibonacci sequences and the spirals formed by employing them have found vast applications in creations and natural phenomena‎. ‎In this study‎, ‎new k-Fibonacci and k-Lucas differential equations are proposed‎. ‎First‎, ‎the k-Fibonacci and k-Lucas sequences are expressed as difference-differential equations‎. ‎Then‎, ‎from the difference-differential equations‎, ‎the associated continuous differential equations are derived‎, ‎which are linear second-order differential equations‎. ‎The initial conditions for the differential equations are written with inspiration from the k-Fibonacci and k-Lucas sequences‎. ‎The solutions‎, ‎which are new spirals‎, ‎are expressed in polar form‎. ‎The spirals produce approximately the k-Fibonacci and k-Lucas numbers at constant steps of angular displacements‎.

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References

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Mathematics Interdisciplinary Research
Volume 11, Issue 2
In Progress
June 2026
Pages 161-176

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